Implicit Differentiation

In calculus, we often deal with equations where the dependent and independent variables are intertwined in a more complex way, such that it’s not possible (or practical) to solve for one variable in terms of the other. These are called implicit equations. Implicit differentiation is the method used to differentiate these equations with respect to a variable, often using the chain rule in the process.

In simple differentiation, we explicitly have a function like $y=f(x)$. However, implicit differentiation allows us to find derivatives of relationships like $F(x,y)=0$, where $y$ is implicitly defined as a function of $x$. We take the derivative of both sides of the equation with respect to $x$, applying the chain rule wherever necessary.

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1. The Concept of Implicit Differentiation

To understand implicit differentiation, consider the relationship $F(x,y)=0$, where $y$ is an implicit function of $x$. We differentiate each term of the equation with respect to $x$, remembering that $y$ is a function of $x$, so we apply the chain rule when differentiating terms involving $y$.

For instance, if we differentiate $y^2$ with respect to $x$, we apply the chain rule:
$\frac{d}{dx}[y^2]=2y\cdot \frac{dy}{dx}$

Thus, implicit differentiation lets us find $\frac{dy}{dx}$ even when we can’t explicitly solve for $y$ in terms of $x$.

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2. Steps for Implicit Differentiation

1. Differentiate both sides of the equation with respect to $x$.
2. Remember to apply the chain rule when differentiating terms involving $y$, since $y$ is implicitly a function of $x$. This means every time you differentiate a $y$ term, you must multiply by $\frac{dy}{dx}$.
3. Solve for $\frac{dy}{dx}$, the derivative of $y$ with respect to $x$.

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3. Example 1: Implicit Differentiation of a Simple Equation

Problem:
Find $\frac{dy}{dx}$ if $x^2+y^2=25$.

Answer:

Step 1: Given Data:
The equation is $x^2+y^2=25$.

Step 2: Solution:
Differentiate both sides of the equation with respect to $x$:
$\frac{d}{dx}[x^2+y^2]=\frac{d}{dx}[25]$

On the left-hand side, differentiate $x^2$ normally:
$\frac{d}{dx}[x^2]=2x$

Next, apply the chain rule to $y^2$:
$\frac{d}{dx}[y^2]=2y\cdot \frac{dy}{dx}$

Since the derivative of a constant ($25$) is zero, we have:
$2x+2y\cdot \frac{dy}{dx}=0$

Now, solve for $\frac{dy}{dx}$:
$2y\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{2y}$
$\frac{dy}{dx}=\frac{-x}{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x}{y}$

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4. Example 2: Implicit Differentiation Involving a Product of $x$ and $y$

Problem:
Find $\frac{dy}{dx}$ if $x^{2}y+y^3=7$.

Answer:

Step 1: Given Data:
The equation is $x^{2}y+y^3=7$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{2}y+y^3]=\frac{d}{dx}[7]$

Apply the product rule to $x^{2}y$:
$\frac{d}{dx}[x^{2}y]=x^2\cdot \frac{dy}{dx}+2x\cdot y$

Now differentiate $y^3$ using the chain rule:
$\frac{d}{dx}[y^3]=3y^2\cdot \frac{dy}{dx}$

On the right-hand side, the derivative of $7$ is zero:
$x^2\cdot \frac{dy}{dx}+2x\cdot y+3y^2\cdot \frac{dy}{dx}=0$

Factor out $\frac{dy}{dx}$ from the terms involving it:
$\frac{dy}{dx}(x^2+3y^2)=-2x\cdot y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2x\cdot y}{x^2+3y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x\cdot y}{x^2+3y^2}$

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5. Example 3: Implicit Differentiation with Trigonometric Functions

Problem:
Find $\frac{dy}{dx}$ if $\sin (xy)=x+y$.

Answer:

Step 1: Given Data:
The equation is $\sin (xy)=x+y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sin (xy)]=\frac{d}{dx}[x+y]$

On the left-hand side, apply the chain rule to differentiate $\sin (xy)$:
$\frac{d}{dx}[\sin (xy)]=\cos (xy)\cdot \frac{d}{dx}[xy]$

Now apply the product rule to $xy$:
$\frac{d}{dx}[xy]=x\cdot \frac{dy}{dx}+y$

So, the derivative of $\sin (xy)$ is:
$\cos (xy)\cdot (x\cdot \frac{dy}{dx}+y)$

On the right-hand side, differentiate $x+y$:
$\frac{d}{dx}[x+y]=1+\frac{dy}{dx}$

Now substitute into the equation:
$\cos (xy)\cdot (x\cdot \frac{dy}{dx}+y)=1+\frac{dy}{dx}$

Distribute $\cos (xy)$:
$\cos (xy)\cdot x\cdot \frac{dy}{dx}+\cos (xy)\cdot y=1+\frac{dy}{dx}$

Now collect all terms involving $\frac{dy}{dx}$ on one side:
$\cos (xy)\cdot x\cdot \frac{dy}{dx}–\frac{dy}{dx}=1–\cos (xy)\cdot y$

Factor out $\frac{dy}{dx}$:
$\frac{dy}{dx}(\cos (xy)\cdot x–1)=1–\cos (xy)\cdot y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{1–\cos (xy)\cdot y}{\cos (xy)\cdot x–1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{1–\cos (xy)\cdot y}{\cos (xy)\cdot x–1}$

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6. Implicit Differentiation and Second Derivatives

Implicit differentiation can also be used to find second derivatives. After finding the first derivative $\frac{dy}{dx}$, we can differentiate it again to find $\frac{d^{2}y}{d{x}^2}$.

Example 4: Finding the Second Derivative

Problem:
Find $\frac{d^{2}y}{d{x}^2}$ if $x^2+y^2=25$.

Answer:

Step 1: Given Data:
The equation is $x^2+y^2=25$.

Step 2: Solution:
We already found that the first derivative is:
$\frac{dy}{dx}=\frac{-x}{y}$

Now, differentiate $\frac{dy}{dx}$ with respect to $x$ to find the second derivative:
$\frac{d}{dx}\left(\frac{-x}{y}\right)$

Apply the quotient rule:
$\frac{d}{dx}\left(\frac{-x}{y}\right)=\frac{(-y\cdot 1–(-x)\cdot \frac{dy}{dx})}{y^2}$

Substitute $\frac{dy}{dx}=\frac{-x}{y}$:
$=\frac{-y+x\cdot \frac{-x}{y}}{y^2}$
$=\frac{-y–\frac{x^2}{y}}{y^2}$
$=\frac{-y^2–x^2}{y^3}$

Step 3: Final Answer:
$\frac{d^{2}y}{d{x}^2}=\frac{-y^2–x^2}{y^3}$

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Conclusion

Implicit differentiation is a powerful technique that allows us to find the derivative of a dependent variable even when it is not explicitly solved in terms of the independent variable. It’s especially useful in cases involving products of variables and complex relationships where standard differentiation methods don’t apply directly.

In optimization problems, physics, and engineering, implicit differentiation is frequently used, making it an indispensable tool in calculus. Whether solving for the slope of a tangent line, optimizing a system, or working with implicit functions, mastering implicit differentiation is crucial for tackling complex calculus problems.

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Question And Answer Library

Example 1: Implicit Differentiation of a Circle Equation

Problem:
Find $\frac{dy}{dx}$ if $x^2+y^2=16$.

Answer:
Step 1: Given Data:
The equation is $x^2+y^2=16$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+y^2]=\frac{d}{dx}[16]$

On the left-hand side, differentiate:
$\frac{d}{dx}[x^2]+\frac{d}{dx}[y^2]=2x+2y\cdot \frac{dy}{dx}$

On the right-hand side, the derivative of a constant is zero:
$2x+2y\cdot \frac{dy}{dx}=0$

Now, solve for $\frac{dy}{dx}$:
$2y\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{2y}$
$\frac{dy}{dx}=\frac{-x}{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x}{y}$

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Example 2: Implicit Differentiation of an Ellipse Equation

Problem:
Find $\frac{dy}{dx}$ if $\frac{x^2}{4}+\frac{y^2}{9}=1$.

Answer:
Step 1: Given Data:
The equation is $\frac{x^2}{4}+\frac{y^2}{9}=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}\left[\frac{x^2}{4}\right]+\frac{d}{dx}\left[\frac{y^2}{9}\right]=\frac{d}{dx}[1]$

Differentiate:
$\frac{1}{4}(2x)+\frac{1}{9}(2y\cdot \frac{dy}{dx})=0$
$\frac{x}{2}+\frac{2y}{9}\cdot \frac{dy}{dx}=0$

Now, solve for $\frac{dy}{dx}$:
$\frac{2y}{9}\cdot \frac{dy}{dx}=-\frac{x}{2}$
$\frac{dy}{dx}=-\frac{x}{2}\cdot \frac{9}{2y}$
$\frac{dy}{dx}=-\frac{9x}{4y}$

Step 3: Final Answer:
$\frac{dy}{dx}=-\frac{9x}{4y}$

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Example 3: Implicit Differentiation of a Hyperbola Equation

Problem:
Find $\frac{dy}{dx}$ if $\frac{x^2}{9}–\frac{y^2}{16}=1$.

Answer:
Step 1: Given Data:
The equation is $\frac{x^2}{9}–\frac{y^2}{16}=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}\left[\frac{x^2}{9}\right]–\frac{d}{dx}\left[\frac{y^2}{16}\right]=\frac{d}{dx}[1]$

Differentiate:
$\frac{1}{9}(2x)–\frac{1}{16}(2y\cdot \frac{dy}{dx})=0$
$\frac{2x}{9}–\frac{y}{8}\cdot \frac{dy}{dx}=0$

Now, solve for $\frac{dy}{dx}$:
$\frac{y}{8}\cdot \frac{dy}{dx}=\frac{2x}{9}$
$\frac{dy}{dx}=\frac{2x}{9}\cdot \frac{8}{y}$
$\frac{dy}{dx}=\frac{16x}{9y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{16x}{9y}$

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Example 4: Implicit Differentiation of a Polynomial

Problem:
Find $\frac{dy}{dx}$ if $x^3+y^3=9$.

Answer:
Step 1: Given Data:
The equation is $x^3+y^3=9$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+y^3]=\frac{d}{dx}[9]$

Differentiate:
$3x^2+3y^2\cdot \frac{dy}{dx}=0$

Now, solve for $\frac{dy}{dx}$:
$3y^2\cdot \frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=\frac{-3x^2}{3y^2}$
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

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Example 5: Implicit Differentiation of a Cubic Equation

Problem:
Find $\frac{dy}{dx}$ if $x^4+y^4=16$.

Answer:
Step 1: Given Data:
The equation is $x^4+y^4=16$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^4+y^4]=\frac{d}{dx}[16]$

Differentiate:
$4x^3+4y^3\cdot \frac{dy}{dx}=0$

Now, solve for $\frac{dy}{dx}$:
$4y^3\cdot \frac{dy}{dx}=-4x^3$
$\frac{dy}{dx}=\frac{-4x^3}{4y^3}$
$\frac{dy}{dx}=\frac{-x^3}{y^3}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^3}{y^3}$

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Example 6: Implicit Differentiation of a Complex Polynomial

Problem:
Find $\frac{dy}{dx}$ if $x^{2}y+x{y}^2=10$.

Answer:
Step 1: Given Data:
The equation is $x^{2}y+x{y}^2=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{2}y+x{y}^2]=\frac{d}{dx}[10]$

Differentiate using the product rule:
$\frac{d}{dx}[x^{2}y]=x^2\frac{dy}{dx}+2xy$
$\frac{d}{dx}[x{y}^2]=y^2+2xy\frac{dy}{dx}$

Combine these results:
$x^2\frac{dy}{dx}+2xy+y^2+2xy\frac{dy}{dx}=0$

Combine like terms:
$(x^2+2xy)\frac{dy}{dx}=-2xy–y^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2xy–y^2}{x^2+2xy}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2xy–y^2}{x^2+2xy}$

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Example 7: Implicit Differentiation with Radical Functions

Problem:
Find $\frac{dy}{dx}$ if $y^2+\sqrt{x}=4$.

Answer:
Step 1: Given Data:
The equation is $y^2+\sqrt{x}=4$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+\sqrt{x}]=\frac{d}{dx}[4]$

Differentiate:
$2y\cdot \frac{dy}{dx}+\frac{1}{2\sqrt{x}}=0$

Now solve for $\frac{dy}{dx}$:
$2y\cdot \frac{dy}{dx}=-\frac{1}{2\sqrt{x}}$
$\frac{dy}{dx}=\frac{-1}{4y\sqrt{x}}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-1}{4y\sqrt{x}}$

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Example 8: Implicit Differentiation of a Logarithmic Function

Problem:
Find $\frac{dy}{dx}$ if $\ln (xy)=x+y$.

Answer:
Step 1: Given Data:
The equation is $\ln (xy)=x+y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\ln (xy)]=\frac{d}{dx}[x+y]$

Using the chain rule:
$\frac{1}{xy}\cdot (y+x\cdot \frac{dy}{dx})=1+\frac{dy}{dx}$

Now multiply both sides by $xy$:
$y+x\cdot \frac{dy}{dx}=xy+xy\cdot \frac{dy}{dx}$

Rearranging gives:
$x\cdot \frac{dy}{dx}–xy\cdot \frac{dy}{dx}=xy–y$
$(x–xy)\cdot \frac{dy}{dx}=xy–y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{xy–y}{x(1–y)}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y(x–1)}{x(1–y)}$

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Example 9: Implicit Differentiation of Exponential Functions

Problem:
Find $\frac{dy}{dx}$ if $e^{xy}=x+y$.

Answer:
Step 1: Given Data:
The equation is $e^{xy}=x+y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[e^{xy}]=\frac{d}{dx}[x+y]$

Using the chain rule on the left:
$e^{xy}\cdot (y+x\cdot \frac{dy}{dx})=1+\frac{dy}{dx}$

Rearranging gives:
$e^{xy}y+e^{xy}x\cdot \frac{dy}{dx}=1+\frac{dy}{dx}$

Now factor out $\frac{dy}{dx}$:
$e^{xy}x\cdot \frac{dy}{dx}–\frac{dy}{dx}=1–e^{xy}y$
$\frac{dy}{dx}(e^{xy}x–1)=1–e^{xy}y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{1–e^{xy}y}{e^{xy}x–1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{1–e^{xy}y}{e^{xy}x–1}$

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Example 10: Implicit Differentiation in a Complex Relationship

Problem:
Find $\frac{dy}{dx}$ if $x^3+xy+y^3=6$.

Answer:
Step 1: Given Data:
The equation is $x^3+xy+y^3=6$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3]+\frac{d}{dx}[xy]+\frac{d}{dx}[y^3]=\frac{d}{dx}[6]$

Differentiate:
$3x^2+(x\cdot \frac{dy}{dx}+y)+3y^2\cdot \frac{dy}{dx}=0$

Combine terms:
$3x^2+x\cdot \frac{dy}{dx}+y+3y^2\cdot \frac{dy}{dx}=0$

Now, factor out $\frac{dy}{dx}$:
$(x+3y^2)\frac{dy}{dx}=-3x^2–y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-3x^2–y}{x+3y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-3x^2–y}{x+3y^2}$

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Question And Answer Library

Example 11: Implicit Differentiation of a Polynomial Equation

Problem:
Find $\frac{dy}{dx}$ if $x^4+y^4=32$.

Answer:
Step 1: Given Data:
The equation is $x^4+y^4=32$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^4+y^4]=\frac{d}{dx}[32]$

Differentiate:
$4x^3+4y^3\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$4y^3\cdot \frac{dy}{dx}=-4x^3$
$\frac{dy}{dx}=\frac{-4x^3}{4y^3}$
$\frac{dy}{dx}=\frac{-x^3}{y^3}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^3}{y^3}$

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Example 12: Implicit Differentiation of an Exponential Equation

Problem:
Find $\frac{dy}{dx}$ if $e^{x+y}=x^2–y$.

Answer:
Step 1: Given Data:
The equation is $e^{x+y}=x^2–y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[e^{x+y}]=\frac{d}{dx}[x^2–y]$

Using the chain rule on the left:
$e^{x+y}\cdot (1+\frac{dy}{dx})=2x–\frac{dy}{dx}$

Now distribute:
$e^{x+y}+e^{x+y}\cdot \frac{dy}{dx}=2x–\frac{dy}{dx}$

Combine like terms:
$e^{x+y}\cdot \frac{dy}{dx}+\frac{dy}{dx}=2x–e^{x+y}$
$\frac{dy}{dx}(e^{x+y}+1)=2x–e^{x+y}$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{2x–e^{x+y}}{e^{x+y}+1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{2x–e^{x+y}}{e^{x+y}+1}$

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Example 13: Implicit Differentiation of a Logarithmic Function

Problem:
Find $\frac{dy}{dx}$ if $\ln (xy)+x^2=5$.

Answer:
Step 1: Given Data:
The equation is $\ln (xy)+x^2=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\ln (xy)]+\frac{d}{dx}[x^2]=\frac{d}{dx}[5]$

Using the product rule for $\ln (xy)$:
$\frac{1}{xy}\cdot (y+x\frac{dy}{dx})+2x=0$

Now simplify:
$\frac{y+x\frac{dy}{dx}}{xy}+2x=0$

Multiply through by $xy$:
$y+x\frac{dy}{dx}+2x^{2}y=0$

Now solve for $\frac{dy}{dx}$:
$x\frac{dy}{dx}=-y–2x^{2}y$
$\frac{dy}{dx}=\frac{-y(1+2x)}{x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y(1+2x)}{x}$

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Example 14: Implicit Differentiation of a Cubic Equation

Problem:
Find $\frac{dy}{dx}$ if $x^3+y^3=27$.

Answer:
Step 1: Given Data:
The equation is $x^3+y^3=27$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+y^3]=\frac{d}{dx}[27]$

Differentiate:
$3x^2+3y^2\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$3y^2\cdot \frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=\frac{-3x^2}{3y^2}$
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

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Example 15: Implicit Differentiation of a Relation Involving xyxyxy

Problem:
Find $\frac{dy}{dx}$ if $xy+y^2=x^3$.

Answer:
Step 1: Given Data:
The equation is $xy+y^2=x^3$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[xy+y^2]=\frac{d}{dx}[x^3]$

Using the product rule on $xy$:
$x\cdot \frac{dy}{dx}+y+2y\cdot \frac{dy}{dx}=3x^2$

Combine like terms:
$(x+2y)\frac{dy}{dx}+y=3x^2$

Now solve for $\frac{dy}{dx}$:
$(x+2y)\frac{dy}{dx}=3x^2–y$
$\frac{dy}{dx}=\frac{3x^2–y}{x+2y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{3x^2–y}{x+2y}$

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Example 16: Implicit Differentiation of a Quadratic Equation

Problem:
Find $\frac{dy}{dx}$ if $x^2–4y^2=8$.

Answer:
Step 1: Given Data:
The equation is $x^2–4y^2=8$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2–4y^2]=\frac{d}{dx}[8]$

Differentiate:
$2x–8y\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$-8y\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{2x}{8y}$
$\frac{dy}{dx}=\frac{x}{4y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{x}{4y}$

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Example 17: Implicit Differentiation of an Equation Involving Trigonometric Functions

Problem:
Find $\frac{dy}{dx}$ if $\sin (xy)=x^2+y$.

Answer:
Step 1: Given Data:
The equation is $\sin (xy)=x^2+y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sin (xy)]=\frac{d}{dx}[x^2+y]$

Using the chain rule:
$\cos (xy)\cdot (y+x\frac{dy}{dx})=2x+\frac{dy}{dx}$

Now distribute:
$\cos (xy)\cdot y+\cos (xy)\cdot x\frac{dy}{dx}=2x+\frac{dy}{dx}$

Collect all terms involving $\frac{dy}{dx}$:
$\cos (xy)\cdot x\frac{dy}{dx}–\frac{dy}{dx}=2x–\cos (xy)\cdot y$
$\frac{dy}{dx}(\cos (xy)\cdot x–1)=2x–\cos (xy)\cdot y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{2x–\cos (xy)\cdot y}{\cos (xy)\cdot x–1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{2x–\cos (xy)\cdot y}{\cos (xy)\cdot x–1}$

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Example 18: Implicit Differentiation of a Complex Relationship

Problem:
Find $\frac{dy}{dx}$ if $x^{2}y+3x{y}^2=10$.

Answer:
Step 1: Given Data:
The equation is $x^{2}y+3x{y}^2=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{2}y+3x{y}^2]=\frac{d}{dx}[10]$

Using the product rule on $x^{2}y$:
$2xy+x^2\frac{dy}{dx}+3(y^2+2xy\frac{dy}{dx})=0$

Combine terms:
$2xy+3y^2+(x^2+6xy)\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$(x^2+6xy)\frac{dy}{dx}=–(2xy+3y^2)$
$\frac{dy}{dx}=\frac{-(2xy+3y^2)}{x^2+6xy}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-(2xy+3y^2)}{x^2+6xy}$

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Example 19: Implicit Differentiation of a Cubic Function

Problem:
Find $\frac{dy}{dx}$ if $x^3+y^3=3xy$.

Answer:
Step 1: Given Data:
The equation is $x^3+y^3=3xy$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+y^3]=\frac{d}{dx}[3xy]$

Differentiate:
$3x^2+3y^2\cdot \frac{dy}{dx}=3(y+x\cdot \frac{dy}{dx})$

Distributing gives:
$3x^2+3y^2\cdot \frac{dy}{dx}=3y+3x\cdot \frac{dy}{dx}$

Now rearrange:
$3y^2\cdot \frac{dy}{dx}–3x\cdot \frac{dy}{dx}=3y–3x^2$
$(3y^2–3x)\cdot \frac{dy}{dx}=3y–3x^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{3y–3x^2}{3y^2–3x}$
$\frac{dy}{dx}=\frac{y–x^2}{y^2–x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y–x^2}{y^2–x}$

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Example 20: Implicit Differentiation of a Relationship with Square Roots

Problem:
Find $\frac{dy}{dx}$ if $x^2+y^2=16$.

Answer:
Step 1: Given Data:
The equation is $x^2+y^2=16$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+y^2]=\frac{d}{dx}[16]$

Differentiate:
$2x+2y\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$2y\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-x}{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x}{y}$

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Example 21: Implicit Differentiation of a Higher Degree Polynomial

Problem:
Find $\frac{dy}{dx}$ if $x^4+y^4=64$.

Answer:
Step 1: Given Data:
The equation is $x^4+y^4=64$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^4+y^4]=\frac{d}{dx}[64]$

Differentiate:
$4x^3+4y^3\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$4y^3\cdot \frac{dy}{dx}=-4x^3$
$\frac{dy}{dx}=\frac{-x^3}{y^3}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^3}{y^3}$

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Example 22: Implicit Differentiation of a Complex Equation

Problem:
Find $\frac{dy}{dx}$ if $x^2{y}^2+xy=10$.

Answer:
Step 1: Given Data:
The equation is $x^2{y}^2+xy=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2{y}^2+xy]=\frac{d}{dx}[10]$

Differentiate:
$2x{y}^2+x^2\cdot 2y\cdot \frac{dy}{dx}+y+x\cdot \frac{dy}{dx}=0$

Now rearrange:
$2x{y}^2+y+(2x^{2}y+x)\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$(2x^{2}y+x)\frac{dy}{dx}=–(2x{y}^2+y)$
$\frac{dy}{dx}=\frac{-(2x{y}^2+y)}{2x^{2}y+x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-(2x{y}^2+y)}{2x^{2}y+x}$

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Example 23: Implicit Differentiation with a Reciprocal Relationship

Problem:
Find $\frac{dy}{dx}$ if $xy=5$.

Answer:
Step 1: Given Data:
The equation is $xy=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[xy]=\frac{d}{dx}[5]$

Using the product rule:
$y+x\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$x\cdot \frac{dy}{dx}=-y$
$\frac{dy}{dx}=\frac{-y}{x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y}{x}$

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Example 24: Implicit Differentiation with a Cubic Function

Problem:
Find $\frac{dy}{dx}$ if $x^3+y^3=9$.

Answer:
Step 1: Given Data:
The equation is $x^3+y^3=9$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+y^3]=\frac{d}{dx}[9]$

Differentiate:
$3x^2+3y^2\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$3y^2\cdot \frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=\frac{-3x^2}{3y^2}$
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

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Example 25: Implicit Differentiation Involving Exponential Functions

Problem:
Find $\frac{dy}{dx}$ if $e^{xy}=x+y$.

Answer:
Step 1: Given Data:
The equation is $e^{xy}=x+y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[e^{xy}]=\frac{d}{dx}[x+y]$

Using the chain rule:
$e^{xy}\cdot (y+x\cdot \frac{dy}{dx})=1+\frac{dy}{dx}$

Now distribute:
$e^{xy}y+e^{xy}x\frac{dy}{dx}=1+\frac{dy}{dx}$

Now collect terms involving $\frac{dy}{dx}$:
$e^{xy}x\frac{dy}{dx}–\frac{dy}{dx}=1–e^{xy}y$
$\frac{dy}{dx}(e^{xy}x–1)=1–e^{xy}y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{1–e^{xy}y}{e^{xy}x–1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{1–e^{xy}y}{e^{xy}x–1}$

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Example 26: Implicit Differentiation of an Equation with a Square Root

Problem:
Find $\frac{dy}{dx}$ if $\sqrt{x^2+y^2}=10$.

Answer:
Step 1: Given Data:
The equation is $\sqrt{x^2+y^2}=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sqrt{x^2+y^2}]=\frac{d}{dx}[10]$

Using the chain rule:
$\frac{1}{2\sqrt{x^2+y^2}}\cdot (2x+2y\cdot \frac{dy}{dx})=0$

Multiply both sides by $2\sqrt{x^2+y^2}$:
$2x+2y\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$2y\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{2y}$
$\frac{dy}{dx}=\frac{-x}{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x}{y}$

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Example 27: Implicit Differentiation of a Logarithmic Relationship

Problem:
Find $\frac{dy}{dx}$ if $\ln (y)+x^2=4$.

Answer:
Step 1: Given Data:
The equation is $\ln (y)+x^2=4$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\ln (y)]+\frac{d}{dx}[x^2]=\frac{d}{dx}[4]$

Using the chain rule for $\ln (y)$:
$\frac{1}{y}\cdot \frac{dy}{dx}+2x=0$

Now solve for $\frac{dy}{dx}$:
$\frac{1}{y}\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=-2xy$

Step 3: Final Answer:
$\frac{dy}{dx}=-2xy$

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Example 28: Implicit Differentiation with a Quadratic Relationship

Problem:
Find $\frac{dy}{dx}$ if $x^2+3y^2=12$.

Answer:
Step 1: Given Data:
The equation is $x^2+3y^2=12$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+3y^2]=\frac{d}{dx}[12]$

Differentiate:
$2x+6y\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$6y\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{6y}$
$\frac{dy}{dx}=\frac{-x}{3y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x}{3y}$

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Example 29: Implicit Differentiation of a Relationship with Mixed Variables

Problem:
Find $\frac{dy}{dx}$ if $xy+x^2{y}^2=10$.

Answer:
Step 1: Given Data:
The equation is $xy+x^2{y}^2=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[xy]+\frac{d}{dx}[x^2{y}^2]=\frac{d}{dx}[10]$

Using the product rule:
$y+x\frac{dy}{dx}+2xy\frac{dy}{dx}+x^2\cdot 2y\frac{dy}{dx}=0$

Now combine terms:
$y+(x+2xy+2x^{2}y)\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$(x+2xy+2x^{2}y)\frac{dy}{dx}=-y$
$\frac{dy}{dx}=\frac{-y}{x+2xy+2x^{2}y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y}{x+2xy+2x^{2}y}$

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Example 30: Implicit Differentiation of a Complex Polynomial

Problem:
Find $\frac{dy}{dx}$ if $x^4+2y^3=16$.

Answer:
Step 1: Given Data:
The equation is $x^4+2y^3=16$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^4+2y^3]=\frac{d}{dx}[16]$

Differentiate:
$4x^3+6y^2\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$6y^2\cdot \frac{dy}{dx}=-4x^3$
$\frac{dy}{dx}=\frac{-4x^3}{6y^2}$
$\frac{dy}{dx}=\frac{-2x^3}{3y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x^3}{3y^2}$

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Example 31: Implicit Differentiation with an Inverse Function

Problem:
Find $\frac{dy}{dx}$ if $x^3+y^3=3xy$.

Answer:
Step 1: Given Data:
The equation is $x^3+y^3=3xy$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+y^3]=\frac{d}{dx}[3xy]$

Differentiate:
$3x^2+3y^2\cdot \frac{dy}{dx}=3y+3x\cdot \frac{dy}{dx}$

Now rearrange:
$3y^2\cdot \frac{dy}{dx}–3x\cdot \frac{dy}{dx}=3y–3x^2$
$(3y^2–3x)\cdot \frac{dy}{dx}=3y–3x^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{3y–3x^2}{3y^2–3x}$
$\frac{dy}{dx}=\frac{y–x^2}{y^2–x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y–x^2}{y^2–x}$

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Example 32: Implicit Differentiation with Higher Powers

Problem:
Find $\frac{dy}{dx}$ if $x^5+y^5=5$.

Answer:
Step 1: Given Data:
The equation is $x^5+y^5=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^5+y^5]=\frac{d}{dx}[5]$

Differentiate:
$5x^4+5y^4\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$5y^4\cdot \frac{dy}{dx}=-5x^4$
$\frac{dy}{dx}=\frac{-5x^4}{5y^4}$
$\frac{dy}{dx}=\frac{-x^4}{y^4}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^4}{y^4}$

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Example 33: Implicit Differentiation of a Relationship with Mixed Exponents

Problem:
Find $\frac{dy}{dx}$ if $x^{3}y+y^3=7$.

Answer:
Step 1: Given Data:
The equation is $x^{3}y+y^3=7$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{3}y+y^3]=\frac{d}{dx}[7]$

Using the product rule on $x^{3}y$:
$3x^{2}y+x^3\frac{dy}{dx}+3y^2\frac{dy}{dx}=0$

Now rearrange:
$x^3\frac{dy}{dx}+3y^2\frac{dy}{dx}=-3x^{2}y$
$(x^3+3y^2)\frac{dy}{dx}=-3x^{2}y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-3x^{2}y}{x^3+3y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-3x^{2}y}{x^3+3y^2}$

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Example 34: Implicit Differentiation Involving a Reciprocal

Problem:
Find $\frac{dy}{dx}$ if $x+\frac{1}{y}=3$.

Answer:
Step 1: Given Data:
The equation is $x+\frac{1}{y}=3$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x+\frac{1}{y}]=\frac{d}{dx}[3]$

Differentiate:
$1–\frac{1}{y^2}\cdot \frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$-\frac{1}{y^2}\frac{dy}{dx}=-1$
$\frac{dy}{dx}=y^2$

Step 3: Final Answer:
$\frac{dy}{dx}=y^2$

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Example 35: Implicit Differentiation of a Cubic Relationship

Problem:
Find $\frac{dy}{dx}$ if $y^3+x^{2}y=9$.

Answer:
Step 1: Given Data:
The equation is $y^3+x^{2}y=9$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^3+x^{2}y]=\frac{d}{dx}[9]$

Differentiate:
$3y^2\cdot \frac{dy}{dx}+(2xy+x^2\frac{dy}{dx})=0$

Now combine like terms:
$(3y^2+x^2)\frac{dy}{dx}=-2xy$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2xy}{3y^2+x^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2xy}{3y^2+x^2}$

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Example 36: Implicit Differentiation of a Relation with Exponential Functions

Problem:
Find $\frac{dy}{dx}$ if $e^y+x^2=10$.

Answer:
Step 1: Given Data:
The equation is $e^y+x^2=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[e^y]+\frac{d}{dx}[x^2]=\frac{d}{dx}[10]$

Using the chain rule:
$e^y\cdot \frac{dy}{dx}+2x=0$

Now solve for $\frac{dy}{dx}$:
$e^y\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{e^y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x}{e^y}$

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Example 37: Implicit Differentiation of an Equation Involving Sine

Problem:
Find $\frac{dy}{dx}$ if $\sin (xy)=y+2$.

Answer:
Step 1: Given Data:
The equation is $\sin (xy)=y+2$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sin (xy)]=\frac{d}{dx}[y+2]$

Using the chain rule:
$\cos (xy)(y+x\frac{dy}{dx})=\frac{dy}{dx}$

Now simplify:
$\cos (xy)y+x\cos (xy)\frac{dy}{dx}=\frac{dy}{dx}$

Now collect terms involving $\frac{dy}{dx}$:
$(x\cos (xy)–1)\frac{dy}{dx}=-\cos (xy)y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-\cos (xy)y}{x\cos (xy)–1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-\cos (xy)y}{x\cos (xy)–1}$

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Example 38: Implicit Differentiation of a Linear Equation

Problem:
Find $\frac{dy}{dx}$ if $x+2y=6$.

Answer:
Step 1: Given Data:
The equation is $x+2y=6$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x+2y]=\frac{d}{dx}[6]$

Differentiate:
$1+2\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$2\frac{dy}{dx}=-1$
$\frac{dy}{dx}=-\frac{1}{2}$

Step 3: Final Answer:
$\frac{dy}{dx}=-\frac{1}{2}$

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Example 39: Implicit Differentiation of an Inverse Function

Problem:
Find $\frac{dy}{dx}$ if $xy–y^2=4$.

Answer:
Step 1: Given Data:
The equation is $xy–y^2=4$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[xy–y^2]=\frac{d}{dx}[4]$

Using the product rule:
$y+x\frac{dy}{dx}–2y\frac{dy}{dx}=0$

Now rearrange:
$(x–2y)\frac{dy}{dx}=-y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-y}{x–2y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y}{x–2y}$

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Example 40: Implicit Differentiation of a Relationship Involving Cosine

Problem:
Find $\frac{dy}{dx}$ if $\cos (x+y)=x–y$.

Answer:
Step 1: Given Data:
The equation is $\cos (x+y)=x–y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\cos (x+y)]=\frac{d}{dx}[x–y]$

Using the chain rule:
$-\sin (x+y)(1+\frac{dy}{dx})=1–\frac{dy}{dx}$

Now distribute:
$-\sin (x+y)–\sin (x+y)\frac{dy}{dx}=1–\frac{dy}{dx}$

Now rearrange:
$-\sin (x+y)\frac{dy}{dx}+\frac{dy}{dx}=1+\sin (x+y)$
$\frac{dy}{dx}(-\sin (x+y)+1)=1+\sin (x+y)$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{1+\sin (x+y)}{1–\sin (x+y)}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{1+\sin (x+y)}{1–\sin (x+y)}$

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Example 41: Implicit Differentiation of a Relationship with Exponential and Polynomial Terms

Problem:
Find $\frac{dy}{dx}$ if $y^2+e^x=x^3+2$.

Answer:
Step 1: Given Data:
The equation is $y^2+e^x=x^3+2$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+e^x]=\frac{d}{dx}[x^3+2]$

Differentiate:
$2y\cdot \frac{dy}{dx}+e^x=3x^2$

Now solve for $\frac{dy}{dx}$:
$2y\cdot \frac{dy}{dx}=3x^2–e^x$
$\frac{dy}{dx}=\frac{3x^2–e^x}{2y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{3x^2–e^x}{2y}$

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Example 42: Implicit Differentiation of a Relationship with Roots

Problem:
Find $\frac{dy}{dx}$ if $\sqrt{x+y}=x–y$.

Answer:
Step 1: Given Data:
The equation is $\sqrt{x+y}=x–y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sqrt{x+y}]=\frac{d}{dx}[x–y]$

Using the chain rule:
$\frac{1}{2\sqrt{x+y}}(1+\frac{dy}{dx})=1–\frac{dy}{dx}$

Now multiply by $2\sqrt{x+y}$:
$1+\frac{dy}{dx}=2\sqrt{x+y}(1–\frac{dy}{dx})$

Now distribute:
$1+\frac{dy}{dx}=2\sqrt{x+y}–2\sqrt{x+y}\frac{dy}{dx}$

Now combine terms:
$\frac{dy}{dx}+2\sqrt{x+y}\frac{dy}{dx}=2\sqrt{x+y}–1$
$(1+2\sqrt{x+y})\frac{dy}{dx}=2\sqrt{x+y}–1$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{2\sqrt{x+y}–1}{1+2\sqrt{x+y}}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{2\sqrt{x+y}–1}{1+2\sqrt{x+y}}$

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Example 43: Implicit Differentiation with a Polynomial and Exponential Function

Problem:
Find $\frac{dy}{dx}$ if $x^3+y^2=e^y$.

Answer:
Step 1: Given Data:
The equation is $x^3+y^2=e^y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+y^2]=\frac{d}{dx}[e^y]$

Differentiate:
$3x^2+2y\cdot \frac{dy}{dx}=e^y\cdot \frac{dy}{dx}$

Now rearrange:
$2y\cdot \frac{dy}{dx}–e^y\cdot \frac{dy}{dx}=-3x^2$
$(2y–e^y)\cdot \frac{dy}{dx}=-3x^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-3x^2}{2y–e^y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-3x^2}{2y–e^y}$

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Example 44: Implicit Differentiation of an Equation with Trigonometric Functions

Problem:
Find $\frac{dy}{dx}$ if $\tan (xy)=x+y$.

Answer:
Step 1: Given Data:
The equation is $\tan (xy)=x+y$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\tan (xy)]=\frac{d}{dx}[x+y]$

Using the chain rule:
${\sec }^2(xy)(y+x\frac{dy}{dx})=1+\frac{dy}{dx}$

Now distribute:
${\sec }^2(xy)y+{\sec }^2(xy)x\frac{dy}{dx}=1+\frac{dy}{dx}$

Now collect all terms involving $\frac{dy}{dx}$:
${\sec }^2(xy)x\frac{dy}{dx}–\frac{dy}{dx}=1–{\sec }^2(xy)y$
$\frac{dy}{dx}({\sec }^2(xy)x–1)=1–{\sec }^2(xy)y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{1–{\sec }^2(xy)y}{{\sec }^2(xy)x–1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{1–{\sec }^2(xy)y}{{\sec }^2(xy)x–1}$

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Example 45: Implicit Differentiation of a Quadratic Equation

Problem:
Find $\frac{dy}{dx}$ if $x^2+4y^2=16$.

Answer:
Step 1: Given Data:
The equation is $x^2+4y^2=16$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+4y^2]=\frac{d}{dx}[16]$

Differentiate:
$2x+8y\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$8y\frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{8y}$
$\frac{dy}{dx}=\frac{-x}{4y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x}{4y}$

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Example 46: Implicit Differentiation of a Complex Relationship

Problem:
Find $\frac{dy}{dx}$ if $x^{2}y+2y^2=10$.

Answer:
Step 1: Given Data:
The equation is $x^{2}y+2y^2=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{2}y+2y^2]=\frac{d}{dx}[10]$

Using the product rule on $x^{2}y$:
$2xy+x^2\frac{dy}{dx}+4y\frac{dy}{dx}=0$

Combine terms:
$(x^2+4y)\frac{dy}{dx}=-2xy$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2xy}{x^2+4y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2xy}{x^2+4y}$

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Example 47: Implicit Differentiation of an Equation with a Root

Problem:
Find $\frac{dy}{dx}$ if $\sqrt{xy}=x+1$.

Answer:
Step 1: Given Data:
The equation is $\sqrt{xy}=x+1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sqrt{xy}]=\frac{d}{dx}[x+1]$

Using the chain rule:
$\frac{1}{2\sqrt{xy}}(y+x\frac{dy}{dx})=1$

Now multiply by $2\sqrt{xy}$:
$y+x\frac{dy}{dx}=2\sqrt{xy}$

Now solve for $\frac{dy}{dx}$:
$x\frac{dy}{dx}=2\sqrt{xy}–y$
$\frac{dy}{dx}=\frac{2\sqrt{xy}–y}{x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{2\sqrt{xy}–y}{x}$

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Example 48: Implicit Differentiation of an Equation with a Mixed Relationship

Problem:
Find $\frac{dy}{dx}$ if $x^2+y^2+xy=7$.

Answer:
Step 1: Given Data:
The equation is $x^2+y^2+xy=7$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+y^2+xy]=\frac{d}{dx}[7]$

Differentiate:
$2x+2y\frac{dy}{dx}+(x\frac{dy}{dx}+y)=0$

Combine like terms:
$(2y+x)\frac{dy}{dx}=-2x–y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2x–y}{2y+x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x–y}{2y+x}$

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Example 49: Implicit Differentiation of a Complex Relationship

Problem:
Find $\frac{dy}{dx}$ if $y^2+x{y}^2=12$.

Answer:
Step 1: Given Data:
The equation is $y^2+x{y}^2=12$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+x{y}^2]=\frac{d}{dx}[12]$

Differentiate:
$2y\frac{dy}{dx}+(y^2+2xy\frac{dy}{dx})=0$

Combine terms:
$(2y+2xy)\frac{dy}{dx}=-y^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-y^2}{2y+2xy}$
$\frac{dy}{dx}=\frac{-y^2}{2(y+xy)}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y^2}{2(y+xy)}$

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Example 50: Implicit Differentiation of a Quadratic Relationship

Problem:
Find $\frac{dy}{dx}$ if $x^2+y^2–4xy=1$.

Answer:
Step 1: Given Data:
The equation is $x^2+y^2–4xy=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+y^2–4xy]=\frac{d}{dx}[1]$

Differentiate:
$2x+2y\frac{dy}{dx}–4(y+x\frac{dy}{dx})=0$

Now combine terms:
$2x+2y\frac{dy}{dx}–4y–4x\frac{dy}{dx}=0$
$(2y–4x)\frac{dy}{dx}=4y–2x$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{4y–2x}{2y–4x}$
$\frac{dy}{dx}=\frac{2(2y–x)}{2y–4x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{2(2y–x)}{2y–4x}$

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Example 51: Implicit Differentiation with Mixed Functions

Problem:
Find $\frac{dy}{dx}$ if $x^{3}y+\ln (y)=5$.

Answer:
Step 1: Given Data:
The equation is $x^{3}y+\ln (y)=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{3}y+\ln (y)]=\frac{d}{dx}[5]$

Using the product rule on $x^{3}y$:
$3x^{2}y+x^3\frac{dy}{dx}+\frac{1}{y}\frac{dy}{dx}=0$

Now collect terms:
$x^3\frac{dy}{dx}+\frac{1}{y}\frac{dy}{dx}=-3x^{2}y$
$(x^3+\frac{1}{y})\frac{dy}{dx}=-3x^{2}y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-3x^{2}y}{x^3+\frac{1}{y}}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-3x^{2}y}{x^3+\frac{1}{y}}$

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Example 52: Implicit Differentiation of an Inverse Function

Problem:
Find $\frac{dy}{dx}$ if $\sin (x+y)=y–x$.

Answer:
Step 1: Given Data:
The equation is $\sin (x+y)=y–x$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sin (x+y)]=\frac{d}{dx}[y–x]$

Using the chain rule:
$\cos (x+y)(1+\frac{dy}{dx})=\frac{dy}{dx}–1$

Now distribute:
$\cos (x+y)+\cos (x+y)\frac{dy}{dx}=\frac{dy}{dx}–1$

Now collect terms:
$\cos (x+y)\frac{dy}{dx}–\frac{dy}{dx}=-1–\cos (x+y)$
$\frac{dy}{dx}(\cos (x+y)–1)=-1–\cos (x+y)$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-1–\cos (x+y)}{\cos (x+y)–1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-1–\cos (x+y)}{\cos (x+y)–1}$

---

Example 53: Implicit Differentiation of an Equation with Two Variables

Problem:
Find $\frac{dy}{dx}$ if $y^2+xy=3$.

Answer:
Step 1: Given Data:
The equation is $y^2+xy=3$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+xy]=\frac{d}{dx}[3]$

Differentiate:
$2y\frac{dy}{dx}+(y+x\frac{dy}{dx})=0$

Now combine terms:
$2y\frac{dy}{dx}+y+x\frac{dy}{dx}=0$
$(2y+x)\frac{dy}{dx}=-y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-y}{2y+x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y}{2y+x}$

---

Example 54: Implicit Differentiation with a Fractional Relationship

Problem:
Find $\frac{dy}{dx}$ if $\frac{y}{x}+x^2=1$.

Answer:
Step 1: Given Data:
The equation is $\frac{y}{x}+x^2=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\frac{y}{x}+x^2]=\frac{d}{dx}[1]$

Using the quotient rule:
$\frac{1\cdot \frac{dy}{dx}–y\cdot 1}{x^2}+2x=0$

Multiply by $x^2$:
$\frac{dy}{dx}–y+2x^3=0$
$\frac{dy}{dx}=y–2x^2$

Step 3: Final Answer:
$\frac{dy}{dx}=y–2x^2$

---

Example 55: Implicit Differentiation with Mixed Powers

Problem:
Find $\frac{dy}{dx}$ if $x^2{y}^3+y=5$.

Answer:
Step 1: Given Data:
The equation is $x^2{y}^3+y=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2{y}^3+y]=\frac{d}{dx}[5]$

Using the product rule:
$2x{y}^3+x^2\cdot 3y^2\frac{dy}{dx}+\frac{dy}{dx}=0$

Now combine terms:
$3x^2{y}^2\frac{dy}{dx}+\frac{dy}{dx}=-2x{y}^3$
$(3x^2{y}^2+1)\frac{dy}{dx}=-2x{y}^3$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2x{y}^3}{3x^2{y}^2+1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x{y}^3}{3x^2{y}^2+1}$

---

Example 56: Implicit Differentiation of a Cubic Polynomial

Problem:
Find $\frac{dy}{dx}$ if $y^3+3xy=7$.

Answer:
Step 1: Given Data:
The equation is $y^3+3xy=7$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^3+3xy]=\frac{d}{dx}[7]$

Differentiate:
$3y^2\frac{dy}{dx}+3(y+x\frac{dy}{dx})=0$

Now combine terms:
$3y^2\frac{dy}{dx}+3y+3x\frac{dy}{dx}=0$
$(3y^2+3x)\frac{dy}{dx}=-3y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-3y}{3y^2+3x}$
$\frac{dy}{dx}=\frac{-y}{y^2+x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y}{y^2+x}$

---

Example 57: Implicit Differentiation of a Complex Relationship

Problem:
Find $\frac{dy}{dx}$ if $y^2+x^{2}y=20$.

Answer:
Step 1: Given Data:
The equation is $y^2+x^{2}y=20$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+x^{2}y]=\frac{d}{dx}[20]$

Differentiate:
$2y\frac{dy}{dx}+(2xy+x^2\frac{dy}{dx})=0$

Now combine terms:
$2y\frac{dy}{dx}+2xy+x^2\frac{dy}{dx}=0$
$(2y+x^2)\frac{dy}{dx}=-2xy$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2xy}{2y+x^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2xy}{2y+x^2}$

---

Example 58: Implicit Differentiation of a Rational Function

Problem:
Find $\frac{dy}{dx}$ if $\frac{y}{x}+x^2=5$.

Answer:
Step 1: Given Data:
The equation is $\frac{y}{x}+x^2=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\frac{y}{x}+x^2]=\frac{d}{dx}[5]$

Using the quotient rule:
$\frac{x\frac{dy}{dx}–y}{x^2}+2x=0$

Multiply through by $x^2$:
$x\frac{dy}{dx}–y+2x^3=0$
$x\frac{dy}{dx}=y–2x^3$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{y–2x^3}{x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y–2x^3}{x}$

---

Example 59: Implicit Differentiation of a Quadratic Equation

Problem:
Find $\frac{dy}{dx}$ if $2x^2+3y^2=8$.

Answer:
Step 1: Given Data:
The equation is $2x^2+3y^2=8$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[2x^2+3y^2]=\frac{d}{dx}[8]$

Differentiate:
$4x+6y\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$6y\frac{dy}{dx}=-4x$
$\frac{dy}{dx}=\frac{-4x}{6y}$
$\frac{dy}{dx}=\frac{-2x}{3y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x}{3y}$

---

Example 60: Implicit Differentiation with Mixed Terms

Problem:
Find $\frac{dy}{dx}$ if $x^{2}y+4x{y}^2=1$.

Answer:
Step 1: Given Data:
The equation is $x^{2}y+4x{y}^2=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{2}y+4x{y}^2]=\frac{d}{dx}[1]$

Using the product rule on $x^{2}y$:
$2xy+x^2\frac{dy}{dx}+4(y^2+2xy\frac{dy}{dx})=0$

Combine terms:
$2xy+4y^2+(x^2+8xy)\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$(x^2+8xy)\frac{dy}{dx}=-2xy–4y^2$
$\frac{dy}{dx}=\frac{-2xy–4y^2}{x^2+8xy}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2xy–4y^2}{x^2+8xy}$

---

Example 61: Implicit Differentiation of a Logarithmic Equation

Problem:
Find $\frac{dy}{dx}$ if $\ln (y)+x^3=4$.

Answer:
Step 1: Given Data:
The equation is $\ln (y)+x^3=4$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\ln (y)]+\frac{d}{dx}[x^3]=\frac{d}{dx}[4]$

Using the chain rule:
$\frac{1}{y}\frac{dy}{dx}+3x^2=0$

Now solve for $\frac{dy}{dx}$:
$\frac{1}{y}\frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=-3x^{2}y$

Step 3: Final Answer:
$\frac{dy}{dx}=-3x^{2}y$

---

Example 62: Implicit Differentiation with a Mixed Equation

Problem:
Find $\frac{dy}{dx}$ if $x^{3}y+y^3=10$.

Answer:
Step 1: Given Data:
The equation is $x^{3}y+y^3=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{3}y+y^3]=\frac{d}{dx}[10]$

Using the product rule:
$3x^{2}y+x^3\frac{dy}{dx}+3y^2\frac{dy}{dx}=0$

Now collect terms:
$(x^3+3y^2)\frac{dy}{dx}=-3x^{2}y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-3x^{2}y}{x^3+3y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-3x^{2}y}{x^3+3y^2}$

---

Example 63: Implicit Differentiation of an Equation with Square Roots

Problem:
Find $\frac{dy}{dx}$ if $\sqrt{y}+x^2=5$.

Answer:
Step 1: Given Data:
The equation is $\sqrt{y}+x^2=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sqrt{y}]+\frac{d}{dx}[x^2]=\frac{d}{dx}[5]$

Using the chain rule:
$\frac{1}{2\sqrt{y}}\frac{dy}{dx}+2x=0$

Now solve for $\frac{dy}{dx}$:
$\frac{1}{2\sqrt{y}}\frac{dy}{dx}=-2x$
$\frac{dy}{dx}=-4x\sqrt{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=-4x\sqrt{y}$

---

Example 64: Implicit Differentiation with a Rational Equation

Problem:
Find $\frac{dy}{dx}$ if $\frac{y}{x^2}+y^2=3$.

Answer:
Step 1: Given Data:
The equation is $\frac{y}{x^2}+y^2=3$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\frac{y}{x^2}+y^2]=\frac{d}{dx}[3]$

Using the quotient rule on $\frac{y}{x^2}$:
$\frac{x^2\frac{dy}{dx}–2y}{x^4}+2y\frac{dy}{dx}=0$

Multiply through by $x^4$:
$x^2\frac{dy}{dx}–2y+2y{x}^2\frac{dy}{dx}=0$

Now combine terms:
$(x^2+2y{x}^2)\frac{dy}{dx}=2y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{2y}{x^2+2y{x}^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{2y}{x^2+2y{x}^2}$

---

Example 65: Implicit Differentiation with Multiple Variables

Problem:
Find $\frac{dy}{dx}$ if $y+x^2{y}^2=5$.

Answer:
Step 1: Given Data:
The equation is $y+x^2{y}^2=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y+x^2{y}^2]=\frac{d}{dx}[5]$

Using the product rule:
$\frac{dy}{dx}+(2x{y}^2+x^2\cdot 2y\frac{dy}{dx})=0$

Now collect terms:
$\frac{dy}{dx}+2x{y}^2+2x^{2}y\frac{dy}{dx}=0$
$(1+2x^{2}y)\frac{dy}{dx}=-2x{y}^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2x{y}^2}{1+2x^{2}y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x{y}^2}{1+2x^{2}y}$

---

Example 66: Implicit Differentiation of an Exponential Relationship

Problem:
Find $\frac{dy}{dx}$ if $e^y+x^3=7$.

Answer:
Step 1: Given Data:
The equation is $e^y+x^3=7$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[e^y]+\frac{d}{dx}[x^3]=\frac{d}{dx}[7]$

Using the chain rule:
$e^y\frac{dy}{dx}+3x^2=0$

Now solve for $\frac{dy}{dx}$:
$e^y\frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=\frac{-3x^2}{e^y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-3x^2}{e^y}$

---

Example 67: Implicit Differentiation with a Root and Linear Term

Problem:
Find $\frac{dy}{dx}$ if $\sqrt{y}+3x=8$.

Answer:
Step 1: Given Data:
The equation is $\sqrt{y}+3x=8$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sqrt{y}]+3=\frac{d}{dx}[8]$

Using the chain rule:
$\frac{1}{2\sqrt{y}}\frac{dy}{dx}+3=0$

Now solve for $\frac{dy}{dx}$:
$\frac{1}{2\sqrt{y}}\frac{dy}{dx}=-3$
$\frac{dy}{dx}=-6\sqrt{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=-6\sqrt{y}$

---

Example 68: Implicit Differentiation of a Trigonometric Function

Problem:
Find $\frac{dy}{dx}$ if $\tan (y)+x^2=5$.

Answer:
Step 1: Given Data:
The equation is $\tan (y)+x^2=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\tan (y)]+\frac{d}{dx}[x^2]=\frac{d}{dx}[5]$

Using the chain rule:
${\sec }^2(y)\cdot \frac{dy}{dx}+2x=0$

Now solve for $\frac{dy}{dx}$:
${\sec }^2(y)\cdot \frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{{\sec }^2(y)}$
$\frac{dy}{dx}=-2x{\cos }^2(y)$

Step 3: Final Answer:
$\frac{dy}{dx}=-2x{\cos }^2(y)$

---

Example 69: Implicit Differentiation of a Cubic Relationship

Problem:
Find $\frac{dy}{dx}$ if $y^3+2xy=3$.

Answer:
Step 1: Given Data:
The equation is $y^3+2xy=3$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^3+2xy]=\frac{d}{dx}[3]$

Differentiate:
$3y^2\frac{dy}{dx}+2(y+x\frac{dy}{dx})=0$

Now combine terms:
$3y^2\frac{dy}{dx}+2y+2x\frac{dy}{dx}=0$
$(3y^2+2x)\frac{dy}{dx}=-2y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2y}{3y^2+2x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2y}{3y^2+2x}$

---

Example 70: Implicit Differentiation of a Logarithmic Function

Problem:
Find $\frac{dy}{dx}$ if $y+\ln (x)=3$.

Answer:
Step 1: Given Data:
The equation is $y+\ln (x)=3$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y]+\frac{d}{dx}[\ln (x)]=\frac{d}{dx}[3]$

Differentiate:
$\frac{dy}{dx}+\frac{1}{x}=0$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=-\frac{1}{x}$

Step 3: Final Answer:
$\frac{dy}{dx}=-\frac{1}{x}$

---

Example 71: Implicit Differentiation of a Quadratic Function

Problem:
Find $\frac{dy}{dx}$ if $2x^2+y^2=9$.

Answer:
Step 1: Given Data:
The equation is $2x^2+y^2=9$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[2x^2+y^2]=\frac{d}{dx}[9]$

Differentiate:
$4x+2y\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$2y\frac{dy}{dx}=-4x$
$\frac{dy}{dx}=\frac{-4x}{2y}$
$\frac{dy}{dx}=\frac{-2x}{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x}{y}$

---

Example 72: Implicit Differentiation of a Complex Polynomial

Problem:
Find $\frac{dy}{dx}$ if $x^4+y^4=16$.

Answer:
Step 1: Given Data:
The equation is $x^4+y^4=16$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^4+y^4]=\frac{d}{dx}[16]$

Differentiate:
$4x^3+4y^3\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$4y^3\frac{dy}{dx}=-4x^3$
$\frac{dy}{dx}=\frac{-4x^3}{4y^3}$
$\frac{dy}{dx}=\frac{-x^3}{y^3}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^3}{y^3}$

---

Example 73: Implicit Differentiation of a Mixed Relationship

Problem:
Find $\frac{dy}{dx}$ if $x^2+4y^2–xy=0$.

Answer:
Step 1: Given Data:
The equation is $x^2+4y^2–xy=0$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+4y^2–xy]=\frac{d}{dx}[0]$

Differentiate:
$2x+8y\frac{dy}{dx}–(y+x\frac{dy}{dx})=0$

Now collect terms:
$2x+8y\frac{dy}{dx}–y–x\frac{dy}{dx}=0$
$(8y–x)\frac{dy}{dx}=-2x+y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2x+y}{8y–x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x+y}{8y–x}$

---

Example 74: Implicit Differentiation of a Logarithmic Function

Problem:
Find $\frac{dy}{dx}$ if $\ln (y)+x^2=5$.

Answer:
Step 1: Given Data:
The equation is $\ln (y)+x^2=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\ln (y)]+\frac{d}{dx}[x^2]=\frac{d}{dx}[5]$

Using the chain rule:
$\frac{1}{y}\frac{dy}{dx}+2x=0$

Now solve for $\frac{dy}{dx}$:
$\frac{1}{y}\frac{dy}{dx}=-2x$
$\frac{dy}{dx}=-2xy$

Step 3: Final Answer:
$\frac{dy}{dx}=-2xy$

---

Example 75: Implicit Differentiation of a Rational Function

Problem:
Find $\frac{dy}{dx}$ if $\frac{y^2}{x}+x=1$.

Answer:
Step 1: Given Data:
The equation is $\frac{y^2}{x}+x=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}\left[\frac{y^2}{x}\right]+\frac{d}{dx}[x]=\frac{d}{dx}[1]$

Using the quotient rule:
$\frac{x(2y\frac{dy}{dx})–y^2}{x^2}+1=0$

Now multiply through by $x^2$:
$x(2y\frac{dy}{dx})–y^2+x^2=0$

Now rearrange:
$x(2y\frac{dy}{dx})=y^2–x^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{y^2–x^2}{2xy}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y^2–x^2}{2xy}$

---

Example 76: Implicit Differentiation of a Quadratic Function

Problem:
Find $\frac{dy}{dx}$ if $y^2–4xy=8$.

Answer:
Step 1: Given Data:
The equation is $y^2–4xy=8$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2–4xy]=\frac{d}{dx}[8]$

Differentiate:
$2y\frac{dy}{dx}–4(y+x\frac{dy}{dx})=0$

Now combine terms:
$2y\frac{dy}{dx}–4y–4x\frac{dy}{dx}=0$
$(2y–4x)\frac{dy}{dx}=4y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{4y}{2y–4x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{4y}{2y–4x}$

---

Example 77: Implicit Differentiation of a Cubic Function

Problem:
Find $\frac{dy}{dx}$ if $y^3–3xy=5$.

Answer:
Step 1: Given Data:
The equation is $y^3–3xy=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^3–3xy]=\frac{d}{dx}[5]$

Differentiate:
$3y^2\frac{dy}{dx}–(3y+3x\frac{dy}{dx})=0$

Now combine terms:
$(3y^2–3x)\frac{dy}{dx}=3y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{3y}{3y^2–3x}$
$\frac{dy}{dx}=\frac{y}{y^2–x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y}{y^2–x}$

---

Example 78: Implicit Differentiation of an Exponential Equation

Problem:
Find $\frac{dy}{dx}$ if $e^y+y^2=x^2+1$.

Answer:
Step 1: Given Data:
The equation is $e^y+y^2=x^2+1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[e^y+y^2]=\frac{d}{dx}[x^2+1]$

Differentiate:
$e^y\frac{dy}{dx}+2y\frac{dy}{dx}=2x$

Now combine terms:
$(e^y+2y)\frac{dy}{dx}=2x$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{2x}{e^y+2y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{2x}{e^y+2y}$

---

Example 79: Implicit Differentiation of a Mixed Relationship

Problem:
Find $\frac{dy}{dx}$ if $y^2+xy=6$.

Answer:
Step 1: Given Data:
The equation is $y^2+xy=6$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+xy]=\frac{d}{dx}[6]$

Differentiate:
$2y\frac{dy}{dx}+(y+x\frac{dy}{dx})=0$

Now combine terms:
$(2y+x)\frac{dy}{dx}=-y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-y}{2y+x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y}{2y+x}$

---

Example 80: Implicit Differentiation of a Polynomial Equation

Problem:
Find $\frac{dy}{dx}$ if $x^3+3y^3=6$.

Answer:
Step 1: Given Data:
The equation is $x^3+3y^3=6$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+3y^3]=\frac{d}{dx}[6]$

Differentiate:
$3x^2+9y^2\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$9y^2\frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=\frac{-3x^2}{9y^2}$
$\frac{dy}{dx}=\frac{-x^2}{3y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^2}{3y^2}$

---

Example 81: Implicit Differentiation of a Complex Function

Problem:
Find $\frac{dy}{dx}$ if $x^{2}y+y^2=4$.

Answer:
Step 1: Given Data:
The equation is $x^{2}y+y^2=4$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^{2}y+y^2]=\frac{d}{dx}[4]$

Using the product rule on $x^{2}y$:
$2xy+x^2\frac{dy}{dx}+2y\frac{dy}{dx}=0$

Now combine terms:
$(x^2+2y)\frac{dy}{dx}=-2xy$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2xy}{x^2+2y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2xy}{x^2+2y}$

---

Example 82: Implicit Differentiation of an Inverse Function

Problem:
Find $\frac{dy}{dx}$ if $x^2+y^2+2xy=0$.

Answer:
Step 1: Given Data:
The equation is $x^2+y^2+2xy=0$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+y^2+2xy]=\frac{d}{dx}[0]$

Differentiate:
$2x+2y\frac{dy}{dx}+2(y+x\frac{dy}{dx})=0$

Now collect terms:
$2x+2y\frac{dy}{dx}+2y+2x\frac{dy}{dx}=0$
$(2y+2x)\frac{dy}{dx}=-2x–2y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2(x+y)}{2(y+x)}$
$\frac{dy}{dx}=-1$

Step 3: Final Answer:
$\frac{dy}{dx}=-1$

---

Example 83: Implicit Differentiation of a Complex Relationship

Problem:
Find $\frac{dy}{dx}$ if $y^2+2x^3=8$.

Answer:
Step 1: Given Data:
The equation is $y^2+2x^3=8$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+2x^3]=\frac{d}{dx}[8]$

Differentiate:
$2y\frac{dy}{dx}+6x^2=0$

Now solve for $\frac{dy}{dx}$:
$2y\frac{dy}{dx}=-6x^2$
$\frac{dy}{dx}=\frac{-6x^2}{2y}$
$\frac{dy}{dx}=\frac{-3x^2}{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-3x^2}{y}$

---

Example 84: Implicit Differentiation of a Polynomial Relationship

Problem:
Find $\frac{dy}{dx}$ if $y^2+3xy=6$.

Answer:
Step 1: Given Data:
The equation is $y^2+3xy=6$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+3xy]=\frac{d}{dx}[6]$

Differentiate:
$2y\frac{dy}{dx}+3(y+x\frac{dy}{dx})=0$

Now combine terms:
$2y\frac{dy}{dx}+3y+3x\frac{dy}{dx}=0$
$(2y+3x)\frac{dy}{dx}=-3y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-3y}{2y+3x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-3y}{2y+3x}$

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Example 85: Implicit Differentiation of a Mixed Polynomial

Problem:
Find $\frac{dy}{dx}$ if $2x^2+y^2=10$.

Answer:
Step 1: Given Data:
The equation is $2x^2+y^2=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[2x^2+y^2]=\frac{d}{dx}[10]$

Differentiate:
$4x+2y\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$2y\frac{dy}{dx}=-4x$
$\frac{dy}{dx}=\frac{-4x}{2y}$
$\frac{dy}{dx}=\frac{-2x}{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x}{y}$

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Example 86: Implicit Differentiation of a Cubic Polynomial

Problem:
Find $\frac{dy}{dx}$ if $y^3–3x^{2}y=5$.

Answer:
Step 1: Given Data:
The equation is $y^3–3x^{2}y=5$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^3–3x^{2}y]=\frac{d}{dx}[5]$

Differentiate:
$3y^2\frac{dy}{dx}–(6xy+3x^2\frac{dy}{dx})=0$

Now combine terms:
$(3y^2–3x^2)\frac{dy}{dx}=6xy$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{6xy}{3y^2–3x^2}$
$\frac{dy}{dx}=\frac{2xy}{y^2–x^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{2xy}{y^2–x^2}$

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Example 87: Implicit Differentiation of a Relationship Involving Square Roots

Problem:
Find $\frac{dy}{dx}$ if $\sqrt{y}+x^3=4$.

Answer:
Step 1: Given Data:
The equation is $\sqrt{y}+x^3=4$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sqrt{y}]+\frac{d}{dx}[x^3]=\frac{d}{dx}[4]$

Using the chain rule:
$\frac{1}{2\sqrt{y}}\frac{dy}{dx}+3x^2=0$

Now solve for $\frac{dy}{dx}$:
$\frac{1}{2\sqrt{y}}\frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=-6x^2\sqrt{y}$

Step 3: Final Answer:
$\frac{dy}{dx}=-6x^2\sqrt{y}$

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Example 88: Implicit Differentiation of a Relationship with Sine

Problem:
Find $\frac{dy}{dx}$ if $\sin (y)+x^2=1$.

Answer:
Step 1: Given Data:
The equation is $\sin (y)+x^2=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\sin (y)]+\frac{d}{dx}[x^2]=\frac{d}{dx}[1]$

Using the chain rule:
$\cos (y)\frac{dy}{dx}+2x=0$

Now solve for $\frac{dy}{dx}$:
$\cos (y)\frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{\cos (y)}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x}{\cos (y)}$

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Example 89: Implicit Differentiation of a Polynomial Relationship

Problem:
Find $\frac{dy}{dx}$ if $3y^3–4xy=8$.

Answer:
Step 1: Given Data:
The equation is $3y^3–4xy=8$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[3y^3–4xy]=\frac{d}{dx}[8]$

Differentiate:
$9y^2\frac{dy}{dx}–(4y+4x\frac{dy}{dx})=0$

Now combine terms:
$(9y^2–4x)\frac{dy}{dx}=4y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{4y}{9y^2–4x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{4y}{9y^2–4x}$

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Example 90: Implicit Differentiation of a Mixed Relationship

Problem:
Find $\frac{dy}{dx}$ if $2x^{2}y+y^2=3$.

Answer:
Step 1: Given Data:
The equation is $2x^{2}y+y^2=3$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[2x^{2}y+y^2]=\frac{d}{dx}[3]$

Using the product rule on $2x^{2}y$:
$2x^2\frac{dy}{dx}+4xy+2y\frac{dy}{dx}=0$

Now combine terms:
$(2x^2+2y)\frac{dy}{dx}=-4xy$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-4xy}{2x^2+2y}$
$\frac{dy}{dx}=\frac{-2xy}{x^2+y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2xy}{x^2+y}$

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Example 91: Implicit Differentiation of a Quadratic Function

Problem:
Find $\frac{dy}{dx}$ if $y^2+2xy=10$.

Answer:
Step 1: Given Data:
The equation is $y^2+2xy=10$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+2xy]=\frac{d}{dx}[10]$

Differentiate:
$2y\frac{dy}{dx}+2(y+x\frac{dy}{dx})=0$

Now combine terms:
$2y\frac{dy}{dx}+2y+2x\frac{dy}{dx}=0$
$(2y+2x)\frac{dy}{dx}=-2y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2y}{2y+2x}$
$\frac{dy}{dx}=\frac{-y}{y+x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y}{y+x}$

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Example 92: Implicit Differentiation of a Cubic Function

Problem:
Find $\frac{dy}{dx}$ if $y^3+x^3=1$.

Answer:
Step 1: Given Data:
The equation is $y^3+x^3=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^3+x^3]=\frac{d}{dx}[1]$

Differentiate:
$3y^2\frac{dy}{dx}+3x^2=0$

Now solve for $\frac{dy}{dx}$:
$3y^2\frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=\frac{-3x^2}{3y^2}$
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

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Example 93: Implicit Differentiation of an Exponential Relationship

Problem:
Find $\frac{dy}{dx}$ if $e^y+x^2=2$.

Answer:
Step 1: Given Data:
The equation is $e^y+x^2=2$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[e^y]+\frac{d}{dx}[x^2]=\frac{d}{dx}[2]$

Using the chain rule:
$e^y\frac{dy}{dx}+2x=0$

Now solve for $\frac{dy}{dx}$:
$e^y\frac{dy}{dx}=-2x$
$\frac{dy}{dx}=\frac{-2x}{e^y}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x}{e^y}$

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Example 94: Implicit Differentiation of a Mixed Polynomial

Problem:
Find $\frac{dy}{dx}$ if $x^3+y^3=27$.

Answer:
Step 1: Given Data:
The equation is $x^3+y^3=27$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+y^3]=\frac{d}{dx}[27]$

Differentiate:
$3x^2+3y^2\frac{dy}{dx}=0$

Now solve for $\frac{dy}{dx}$:
$3y^2\frac{dy}{dx}=-3x^2$
$\frac{dy}{dx}=\frac{-3x^2}{3y^2}$
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-x^2}{y^2}$

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Example 95: Implicit Differentiation of a Rational Function

Problem:
Find $\frac{dy}{dx}$ if $\frac{y^2}{x}+x=4$.

Answer:
Step 1: Given Data:
The equation is $\frac{y^2}{x}+x=4$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[\frac{y^2}{x}+x]=\frac{d}{dx}[4]$

Using the quotient rule on $\frac{y^2}{x}$:
$\frac{x(2y\frac{dy}{dx})–y^2}{x^2}+1=0$

Now multiply by $x^2$:
$x(2y\frac{dy}{dx})–y^2+x^2=0$

Now rearrange:
$x(2y\frac{dy}{dx})=y^2–x^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{y^2–x^2}{2xy}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y^2–x^2}{2xy}$

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Example 96: Implicit Differentiation of a Complex Function

Problem:
Find $\frac{dy}{dx}$ if $y+\ln (y)=x$.

Answer:
Step 1: Given Data:
The equation is $y+\ln (y)=x$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y]+\frac{d}{dx}[\ln (y)]=\frac{d}{dx}[x]$

Using the chain rule:
$\frac{dy}{dx}+\frac{1}{y}\frac{dy}{dx}=1$

Now combine terms:
$(1+\frac{1}{y})\frac{dy}{dx}=1$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{y}{y+1}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y}{y+1}$

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Example 97: Implicit Differentiation of a Polynomial Relationship

Problem:
Find $\frac{dy}{dx}$ if $y^2+2x^{2}y=12$.

Answer:
Step 1: Given Data:
The equation is $y^2+2x^{2}y=12$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y^2+2x^{2}y]=\frac{d}{dx}[12]$

Using the product rule on $2x^{2}y$:
$2y\frac{dy}{dx}+2y+4xy\frac{dy}{dx}=0$

Now combine terms:
$(2y+4xy)\frac{dy}{dx}=-2y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2y}{2y+4xy}$
$\frac{dy}{dx}=\frac{-y}{y+2xy}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-y}{y+2xy}$

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Example 98: Implicit Differentiation of a Mixed Polynomial

Problem:
Find $\frac{dy}{dx}$ if $x^2+y^3–3xy=0$.

Answer:
Step 1: Given Data:
The equation is $x^2+y^3–3xy=0$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^2+y^3–3xy]=0$

Differentiate:
$2x+3y^2\frac{dy}{dx}–(3y+3x\frac{dy}{dx})=0$

Now combine terms:
$3y^2\frac{dy}{dx}–3x\frac{dy}{dx}=-2x–3y$
$(3y^2–3x)\frac{dy}{dx}=-2x–3y$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{-2x–3y}{3y^2–3x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{-2x–3y}{3y^2–3x}$

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Example 99: Implicit Differentiation of a Logarithmic Relationship

Problem:
Find $\frac{dy}{dx}$ if $y+\ln (x)=1$.

Answer:
Step 1: Given Data:
The equation is $y+\ln (x)=1$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[y]+\frac{d}{dx}[\ln (x)]=\frac{d}{dx}[1]$

Differentiate:
$\frac{dy}{dx}+\frac{1}{x}=0$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=-\frac{1}{x}$

Step 3: Final Answer:
$\frac{dy}{dx}=-\frac{1}{x}$

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Example 100: Implicit Differentiation of a Cubic Relationship

Problem:
Find $\frac{dy}{dx}$ if $x^3+y^3=3xy$.

Answer:
Step 1: Given Data:
The equation is $x^3+y^3=3xy$.

Step 2: Solution:
Differentiate both sides with respect to $x$:
$\frac{d}{dx}[x^3+y^3]=\frac{d}{dx}[3xy]$

Differentiate:
$3x^2+3y^2\frac{dy}{dx}=3(y+x\frac{dy}{dx})$

Now rearrange:
$3y^2\frac{dy}{dx}–3x\frac{dy}{dx}=3y–3x^2$
$(3y^2–3x)\frac{dy}{dx}=3y–3x^2$

Now solve for $\frac{dy}{dx}$:
$\frac{dy}{dx}=\frac{3y–3x^2}{3y^2–3x}$
$\frac{dy}{dx}=\frac{y–x^2}{y^2–x}$

Step 3: Final Answer:
$\frac{dy}{dx}=\frac{y–x^2}{y^2–x}$