Derivatives and Differentiation

In calculus, differentiation is the process of finding the derivative of a function. The derivative of a function measures how the function’s output changes as its input changes. It represents the rate of change or the slope of the tangent line to the function at any given point. Derivatives are foundational in calculus and are used extensively in physics, engineering, economics, and many other fields to model and analyze dynamic systems.

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1. Definition of the Derivative

The derivative of a function $f(x)$ at a point $x$ is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. This is mathematically expressed as:

$f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)–f(x)}{\mathrm{\Delta }x}$

This definition provides a precise way to find the slope of the function at any point.

Interpretation:

• If $f^{\prime }(x)>0$, the function is increasing at $x$.
• If $f^{\prime }(x)<0$, the function is decreasing at $x$.
• If $f^{\prime }(x)=0$, the function has a critical point at $x$ (this could be a local maximum, local minimum, or point of inflection).

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2. Notations for Derivatives

There are several common notations for derivatives:

• Leibniz notation: $\frac{dy}{dx}$ or $\frac{df(x)}{dx}$
• Lagrange notation: $f^{\prime }(x)$
• Newton’s notation: $\dot{y}$ (used primarily in physics for time derivatives)

All of these notations represent the derivative of $y$ with respect to $x$.

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3. Basic Differentiation Rules

To compute derivatives, there are several fundamental rules to follow:

Power Rule:

If $f(x)=x^n$, then $f^{\prime }(x)=n{x}^{n-1}$.

Constant Rule:

If $f(x)=c$, where $c$ is a constant, then $f^{\prime }(x)=0$.

Sum Rule:

If $f(x)=g(x)+h(x)$, then $f^{\prime }(x)=g^{\prime }(x)+h^{\prime }(x)$.

Difference Rule:

If $f(x)=g(x)–h(x)$, then $f^{\prime }(x)=g^{\prime }(x)–h^{\prime }(x)$.

Product Rule:

If $f(x)=g(x)h(x)$, then $f^{\prime }(x)=g^{\prime }(x)h(x)+g(x)h^{\prime }(x)$.

Quotient Rule:

If $f(x)=\frac{g(x)}{h(x)}$, then

$f^{\prime }(x)=\frac{g^{\prime }(x)h(x)–g(x)h^{\prime }(x)}{h(x{)}^2}$.

Chain Rule:

If $f(x)=g(h(x))$, then $f^{\prime }(x)=g^{\prime }(h(x))h^{\prime }(x)$.

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4. Derivatives of Common Functions

There are several common functions whose derivatives are frequently used in calculus:

1. Derivative of $x^n$:
   $\frac{d}{dx}(x^n)=n{x}^{n-1}$
2. Derivative of $\sin (x)$:
   $\frac{d}{dx}(\sin (x))=\cos (x)$
3. Derivative of $\cos (x)$:
   $\frac{d}{dx}(\cos (x))=-\sin (x)$
4. Derivative of $e^x$:
   $\frac{d}{dx}(e^x)=e^x$
5. Derivative of $\ln (x)$:
   $\frac{d}{dx}(\ln (x))=\frac{1}{x}$

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5. Higher-Order Derivatives

The second derivative of a function is the derivative of the derivative, which is written as $f”(x)$ or $\frac{d^{2}y}{d{x}^2}$. It measures the concavity of the function:

• If $f”(x)>0$, the graph of $f(x)$ is concave up.
• If $f”(x)<0$, the graph of $f(x)$ is concave down.

The third derivative is denoted as $f{”}^{\prime }(x)$ or $\frac{d^{3}y}{d{x}^3}$ and so on for higher derivatives.

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6. Applications of Derivatives

Derivatives have a wide range of applications:

• Tangent and normal lines: The derivative at a point gives the slope of the tangent line to the curve at that point.
• Velocity and acceleration: In physics, the derivative of the position function with respect to time gives the velocity, and the second derivative gives the acceleration.
• Optimization: Derivatives are used to find local maxima and minima of functions, which are essential in optimization problems.
• Curve sketching: The derivative provides critical information about the shape of a function, such as where it increases, decreases, and changes concavity.

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Examples of Differentiation

Example 1: Differentiating a Polynomial

Problem:
Find the derivative of $f(x)=3x^2+2x–5$.

Answer:
Step 1: Given Data:
$f(x)=3x^2+2x–5$

Step 2: Solution:
Using the power rule for each term:

$f^{\prime }(x)=\frac{d}{dx}(3x^2)+\frac{d}{dx}(2x)–\frac{d}{dx}(5)$

$=6x+2–0$

Step 3: Final Answer:
$f^{\prime }(x)=6x+2$

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Example 2: Differentiating a Trigonometric Function

Problem:
Find the derivative of $f(x)=\sin (x)+\cos (x)$.

Answer:
Step 1: Given Data:
$f(x)=\sin (x)+\cos (x)$

Step 2: Solution:
Using the known derivatives of sine and cosine:

$f^{\prime }(x)=\frac{d}{dx}(\sin (x))+\frac{d}{dx}(\cos (x))$

$=\cos (x)–\sin (x)$

Step 3: Final Answer:
$f^{\prime }(x)=\cos (x)–\sin (x)$

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Example 3: Using the Product Rule

Problem:
Find the derivative of $f(x)=x^2\sin (x)$.

Answer:
Step 1: Given Data:
$f(x)=x^2\sin (x)$

Step 2: Solution:
Using the product rule:

$f^{\prime }(x)=\frac{d}{dx}(x^2)\sin (x)+x^2\frac{d}{dx}(\sin (x))$

$=2x\sin (x)+x^2\cos (x)$

Step 3: Final Answer:
$f^{\prime }(x)=2x\sin (x)+x^2\cos (x)$

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Example 4: Using the Quotient Rule

Problem:
Find the derivative of $f(x)=\frac{x^2+1}{x}$.

Answer:
Step 1: Given Data:
$f(x)=\frac{x^2+1}{x}$

Step 2: Solution:
Using the quotient rule:

$f^{\prime }(x)=\frac{(2x)(x)–(x^2+1)(1)}{x^2}$

$=\frac{2x^2–x^2–1}{x^2}$

$=\frac{x^2–1}{x^2}$

Step 3: Final Answer:
$f^{\prime }(x)=1–\frac{1}{x^2}$

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Example 5: Using the Chain Rule

Problem:
Find the derivative of $f(x)=(3x+1{)}^4$.

Answer:
Step 1: Given Data:
$f(x)=(3x+1{)}^4$

Step 2: Solution:
Using the chain rule:

$f^{\prime }(x)=4(3x+1{)}^3\cdot \frac{d}{dx}(3x+1)$

$=4(3x+1{)}^3\cdot 3$

$=12(3x+1{)}^3$

Step 3: Final Answer:
$f^{\prime }(x)=12(3x+1{)}^3$

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Conclusion

Differentiation is one of the two fundamental operations in calculus, the other being integration. By finding derivatives, we can understand how functions change, find slopes of curves, and apply these concepts in real-world situations such as motion, optimization, and economic models. Understanding the basic differentiation rules, applying them to solve problems, and recognizing when to use more advanced rules like the product, quotient, and chain rules are essential skills in calculus.

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Question And Answer Library

Example 1: Differentiating a Polynomial

Problem:
Find the derivative of $f(x)=4x^3–2x^2+5x–7$.

Answer:
Step 1: Given Data:
$f(x)=4x^3–2x^2+5x–7$

Step 2: Solution:
Using the power rule for each term:
$f^{\prime }(x)=\frac{d}{dx}(4x^3)–\frac{d}{dx}(2x^2)+\frac{d}{dx}(5x)–\frac{d}{dx}(7)$
$=12x^2–4x+5–0$

Step 3: Final Answer:
$f^{\prime }(x)=12x^2–4x+5$

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Example 2: Differentiating a Trigonometric Function

Problem:
Find the derivative of $f(x)=\tan (x)+{\sec }^2(x)$.

Answer:
Step 1: Given Data:
$f(x)=\tan (x)+{\sec }^2(x)$

Step 2: Solution:
Using the known derivatives:
$f^{\prime }(x)=\frac{d}{dx}(\tan (x))+\frac{d}{dx}({\sec }^2(x))$
$={\sec }^2(x)+2{\sec }^2(x)\tan (x)$

Step 3: Final Answer:
$f^{\prime }(x)={\sec }^2(x)+2{\sec }^2(x)\tan (x)$

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Example 3: Using the Product Rule

Problem:
Find the derivative of $f(x)=x^2\ln (x)$.

Answer:
Step 1: Given Data:
$f(x)=x^2\ln (x)$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=\frac{d}{dx}(x^2)\ln (x)+x^2\frac{d}{dx}(\ln (x))$
$=2x\ln (x)+x^2\cdot \frac{1}{x}$
$=2x\ln (x)+x$

Step 3: Final Answer:
$f^{\prime }(x)=2x\ln (x)+x$

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Example 4: Using the Quotient Rule

Problem:
Find the derivative of $f(x)=\frac{x^2+1}{x^3}$.

Answer:
Step 1: Given Data:
$f(x)=\frac{x^2+1}{x^3}$

Step 2: Solution:
Using the quotient rule:
$f^{\prime }(x)=\frac{(2x)(x^3)–(x^2+1)(3x^2)}{(x^3{)}^2}$
$=\frac{2x^4–(3x^2+3)}{x^6}$
$=\frac{2x^4–3x^2–3}{x^6}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{2x^4–3x^2–3}{x^6}$

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Example 5: Using the Chain Rule

Problem:
Find the derivative of $f(x)=\sqrt{5x^2+3}$.

Answer:
Step 1: Given Data:
$f(x)=\sqrt{5x^2+3}$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{2\sqrt{5x^2+3}}\cdot \frac{d}{dx}(5x^2+3)$
$=\frac{1}{2\sqrt{5x^2+3}}\cdot 10x$
$=\frac{5x}{\sqrt{5x^2+3}}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{5x}{\sqrt{5x^2+3}}$

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Example 6: Derivative of a Constant Function

Problem:
Find the derivative of $f(x)=8$.

Answer:
Step 1: Given Data:
$f(x)=8$

Step 2: Solution:
Using the constant rule:
$f^{\prime }(x)=0$

Step 3: Final Answer:
$f^{\prime }(x)=0$

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Example 7: Implicit Differentiation

Problem:
Differentiate $x^2+y^2=25$ with respect to $x$.

Answer:
Step 1: Given Data:
$x^2+y^2=25$

Step 2: Solution:
Differentiating both sides:
$\frac{d}{dx}(x^2)+\frac{d}{dx}(y^2)=0$
$2x+2y\frac{dy}{dx}=0$

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

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Example 8: Higher Order Derivatives

Problem:
Find the second derivative of $f(x)=x^3+2x^2+x+1$.

Answer:
Step 1: Given Data:
$f(x)=x^3+2x^2+x+1$

Step 2: Solution:
First derivative:
$f^{\prime }(x)=3x^2+4x+1$

Second derivative:
$f”(x)=6x+4$

Step 3: Final Answer:
$f”(x)=6x+4$

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Example 9: Using the Power Rule

Problem:
Find the derivative of $f(x)=7x^4–3x^2+5x–1$.

Answer:
Step 1: Given Data:
$f(x)=7x^4–3x^2+5x–1$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=28x^3–6x+5$

Step 3: Final Answer:
$f^{\prime }(x)=28x^3–6x+5$

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Example 10: Derivative of an Exponential Function

Problem:
Find the derivative of $f(x)=e^{2x}$.

Answer:
Step 1: Given Data:
$f(x)=e^{2x}$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=e^{2x}\cdot \frac{d}{dx}(2x)$
$=2e^{2x}$

Step 3: Final Answer:
$f^{\prime }(x)=2e^{2x}$

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Example 11: Derivative of a Logarithmic Function

Problem:
Find the derivative of $f(x)=\ln (x^2+1)$.

Answer:
Step 1: Given Data:
$f(x)=\ln (x^2+1)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{x^2+1}\cdot \frac{d}{dx}(x^2+1)$
$=\frac{1}{x^2+1}\cdot 2x$
$=\frac{2x}{x^2+1}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{2x}{x^2+1}$

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Example 12: Finding Derivative Using Implicit Differentiation

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

Answer:
Step 1: Given Data:
$x^3+y^3=6xy$

Step 2: Solution:
Differentiating both sides:
$3x^2+3y^2\frac{dy}{dx}=6(y+x\frac{dy}{dx})$

Rearranging gives:
$3y^2\frac{dy}{dx}–6x\frac{dy}{dx}=6y–3x^2$
$(3y^2–6x)\frac{dy}{dx}=6y–3x^2$
Thus,
$\frac{dy}{dx}=\frac{6y–3x^2}{3y^2–6x}$

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

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Example 13: Differentiating a Cubic Function

Problem:
Find the derivative of $f(x)=x^3–3x^2+4x–2$.

Answer:
Step 1: Given Data:
$f(x)=x^3–3x^2+4x–2$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=3x^2–6x+4$

Step 3: Final Answer:
$f^{\prime }(x)=3x^2–6x+4$

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Example 14: Derivative of a Sinusoidal Function

Problem:
Find the derivative of $f(x)=5\sin (2x)$.

Answer:
Step 1: Given Data:
$f(x)=5\sin (2x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=5\cdot \cos (2x)\cdot \frac{d}{dx}(2x)$
$=10\cos (2x)$

Step 3: Final Answer:
$f^{\prime }(x)=10\cos (2x)$

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Example 15: Derivative of a Cosine Function

Problem:
Find the derivative of $f(x)=\cos (3x)$.

Answer:
Step 1: Given Data:
$f(x)=\cos (3x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=-\sin (3x)\cdot \frac{d}{dx}(3x)$
$=-3\sin (3x)$

Step 3: Final Answer:
$f^{\prime }(x)=-3\sin (3x)$

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Example 16: Derivative of a Hyperbolic Function

Problem:
Find the derivative of $f(x)=\sinh (x)+\cosh (x)$.

Answer:
Step 1: Given Data:
$f(x)=\sinh (x)+\cosh (x)$

Step 2: Solution:
Using the known derivatives of hyperbolic functions:
$f^{\prime }(x)=\cosh (x)+\sinh (x)$

Step 3: Final Answer:
$f^{\prime }(x)=\cosh (x)+\sinh (x)$

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Example 17: Implicit Differentiation of a Circle

Problem:
Find $\frac{dy}{dx}$ for the circle equation $x^2+y^2=25$.

Answer:
Step 1: Given Data:
$x^2+y^2=25$

Step 2: Solution:
Differentiating both sides:
$2x+2y\frac{dy}{dx}=0$
$2y\frac{dy}{dx}=-2x$
Thus,
$\frac{dy}{dx}=-\frac{x}{y}$

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

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Example 18: Derivative of an Exponential Function with a Base Other Than e

Problem:
Find the derivative of $f(x)=3^x$.

Answer:
Step 1: Given Data:
$f(x)=3^x$

Step 2: Solution:
Using the derivative of an exponential function:
$f^{\prime }(x)=3^x\ln (3)$

Step 3: Final Answer:
$f^{\prime }(x)=3^x\ln (3)$

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Example 19: Derivative of an Absolute Value Function

Problem:
Find the derivative of $f(x)=|x|$.

Answer:
Step 1: Given Data:
$f(x)=|x|$

Step 2: Solution:
The derivative is defined piecewise:
$f^{\prime }(x)=\{\begin{array}{llll}1& \text{if }x>0-1& \text{if }x<0\text{undefined}& \text{if }x=0\end{array}$

Step 3: Final Answer:
$f^{\prime }(x)=\{\begin{array}{llll}1& \text{if }x>0-1& \text{if }x<0\text{undefined}& \text{if }x=0\end{array}$

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Example 20: Finding Derivative of a Logarithmic Function with a Base Other Than e

Problem:
Find the derivative of $f(x)={\log }_2(x)$.

Answer:
Step 1: Given Data:
$f(x)={\log }_2(x)$

Step 2: Solution:
Using the change of base formula:
$f^{\prime }(x)=\frac{1}{x\ln (2)}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{1}{x\ln (2)}$

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Example 21: Derivative of a Composite Function

Problem:
Find the derivative of $f(x)=\sin (x^2)$.

Answer:
Step 1: Given Data:
$f(x)=\sin (x^2)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\cos (x^2)\cdot \frac{d}{dx}(x^2)$
$=\cos (x^2)\cdot 2x$

Step 3: Final Answer:
$f^{\prime }(x)=2x\cos (x^2)$

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Example 22: Derivative of a Quadratic Function

Problem:
Find the derivative of $f(x)=2x^2–4x+3$.

Answer:
Step 1: Given Data:
$f(x)=2x^2–4x+3$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=4x–4$

Step 3: Final Answer:
$f^{\prime }(x)=4x–4$

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Example 23: Derivative of a Function Defined Piecewise

Problem:
Find the derivative of
$f(x)=\{\begin{array}{lll}{x}^2& \text{if }x<13x+2& \text{if }x\ge 1\end{array}$.

Answer:
Step 1: Given Data:
$f(x)=\{\begin{array}{lll}{x}^2& \text{if }x<13x+2& \text{if }x\ge 1\end{array}$

Step 2: Solution:
For $x<1$:
$f^{\prime }(x)=2x$

For $x\ge 1$:
$f^{\prime }(x)=3$

Step 3: Final Answer:
$f^{\prime }(x)=\{\begin{array}{lll}2x& \text{if }x<13& \text{if }x\ge 1\end{array}$

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Example 24: Higher Order Derivatives

Problem:
Find the second derivative of $f(x)=x^4–6x^3+9x^2$.

Answer:
Step 1: Given Data:
$f(x)=x^4–6x^3+9x^2$

Step 2: Solution:
First derivative:
$f^{\prime }(x)=4x^3–18x^2+18x$

Second derivative:
$f”(x)=12x^2–36x+18$

Step 3: Final Answer:
$f”(x)=12x^2–36x+18$

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Example 25: Using the Implicit Function Theorem

Problem:
Find $\frac{dy}{dx}$ for the equation $x^2+xy+y^2=6$.

Answer:
Step 1: Given Data:
$x^2+xy+y^2=6$

Step 2: Solution:
Differentiating both sides:
$2x+y+x\frac{dy}{dx}+2y\frac{dy}{dx}=0$

Rearranging gives:
$(x+2y)\frac{dy}{dx}=-2x–y$
Thus,
$\frac{dy}{dx}=\frac{-2x–y}{x+2y}$

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

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Example 26: Derivative of a Cubic Function

Problem:
Find the derivative of $f(x)=4x^3–2x^2+x+3$.

Answer:
Step 1: Given Data:
$f(x)=4x^3–2x^2+x+3$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=12x^2–4x+1$

Step 3: Final Answer:
$f^{\prime }(x)=12x^2–4x+1$

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Example 27: Derivative of an Inverse Function

Problem:
Find the derivative of $f(x)={\sin }^{-1}(x)$.

Answer:
Step 1: Given Data:
$f(x)={\sin }^{-1}(x)$

Step 2: Solution:
Using the known derivative:
$f^{\prime }(x)=\frac{1}{\sqrt{1–x^2}}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{1}{\sqrt{1–x^2}}$

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Example 28: Derivative of a Composite Function

Problem:
Find the derivative of $f(x)=\ln (3x^2+2)$.

Answer:
Step 1: Given Data:
$f(x)=\ln (3x^2+2)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{3x^2+2}\cdot \frac{d}{dx}(3x^2+2)$
$=\frac{1}{3x^2+2}\cdot 6x$
$=\frac{6x}{3x^2+2}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{6x}{3x^2+2}$

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Example 29: Derivative of an Exponential Function

Problem:
Find the derivative of $f(x)=7^x$.

Answer:
Step 1: Given Data:
$f(x)=7^x$

Step 2: Solution:
Using the derivative of an exponential function:
$f^{\prime }(x)=7^x\ln (7)$

Step 3: Final Answer:
$f^{\prime }(x)=7^x\ln (7)$

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Example 30: Finding the Derivative of a Logarithmic Function

Problem:
Find the derivative of $f(x)={\log }_3(x)$.

Answer:
Step 1: Given Data:
$f(x)={\log }_3(x)$

Step 2: Solution:
Using the change of base formula:
$f^{\prime }(x)=\frac{1}{x\ln (3)}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{1}{x\ln (3)}$

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Example 31: Derivative of a Rational Function

Problem:
Find the derivative of $f(x)=\frac{1}{x^2+1}$.

Answer:
Step 1: Given Data:
$f(x)=\frac{1}{x^2+1}$

Step 2: Solution:
Using the quotient rule:
$f^{\prime }(x)=\frac{(0)(x^2+1)–(1)(2x)}{(x^2+1{)}^2}$
$=\frac{-2x}{(x^2+1{)}^2}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{-2x}{(x^2+1{)}^2}$

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Example 32: Derivative of a Piecewise Function

Problem:
Find the derivative of
$f(x)=\{\begin{array}{lll}{x}^2& \text{if }x<02x+1& \text{if }x\ge 0\end{array}$.

Answer:
Step 1: Given Data:
$f(x)=\{\begin{array}{lll}{x}^2& \text{if }x<02x+1& \text{if }x\ge 0\end{array}$

Step 2: Solution:
For $x<0$:
$f^{\prime }(x)=2x$

For $x\ge 0$:
$f^{\prime }(x)=2$

Step 3: Final Answer:
$f^{\prime }(x)=\{\begin{array}{lll}2x& \text{if }x<02& \text{if }x\ge 0\end{array}$

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Example 33: Finding the Derivative of a Polynomial

Problem:
Find the derivative of $f(x)=x^5–3x^3+x–8$.

Answer:
Step 1: Given Data:
$f(x)=x^5–3x^3+x–8$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=5x^4–9x^2+1$

Step 3: Final Answer:
$f^{\prime }(x)=5x^4–9x^2+1$

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Example 34: Derivative of an Arc Function

Problem:
Find the derivative of $f(x)=\arctan (x)$.

Answer:
Step 1: Given Data:
$f(x)=\arctan (x)$

Step 2: Solution:
Using the known derivative:
$f^{\prime }(x)=\frac{1}{1+x^2}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{1}{1+x^2}$

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Example 35: Derivative of a Complex Function

Problem:
Find the derivative of $f(x)=e^{x^2}\sin (x)$.

Answer:
Step 1: Given Data:
$f(x)=e^{x^2}\sin (x)$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=\frac{d}{dx}(e^{x^2})\sin (x)+e^{x^2}\frac{d}{dx}(\sin (x))$
$=2x{e}^{x^2}\sin (x)+e^{x^2}\cos (x)$

Step 3: Final Answer:
$f^{\prime }(x)=2x{e}^{x^2}\sin (x)+e^{x^2}\cos (x)$

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Example 36: Derivative of a Composite Trigonometric Function

Problem:
Find the derivative of $f(x)=\cos (2x)+\tan (x)$.

Answer:
Step 1: Given Data:
$f(x)=\cos (2x)+\tan (x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=-\sin (2x)\cdot 2+{\sec }^2(x)$
$=-2\sin (2x)+{\sec }^2(x)$

Step 3: Final Answer:
$f^{\prime }(x)=-2\sin (2x)+{\sec }^2(x)$

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Example 37: Derivative of a Natural Logarithm Function

Problem:
Find the derivative of $f(x)=\ln (x^3+4x)$.

Answer:
Step 1: Given Data:
$f(x)=\ln (x^3+4x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{x^3+4x}\cdot (3x^2+4)$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{3x^2+4}{x^3+4x}$

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Example 38: Derivative of a Power Function

Problem:
Find the derivative of $f(x)=(5x+3{)}^4$.

Answer:
Step 1: Given Data:
$f(x)=(5x+3{)}^4$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=4(5x+3{)}^3\cdot \frac{d}{dx}(5x+3)$
$=4(5x+3{)}^3\cdot 5$
$=20(5x+3{)}^3$

Step 3: Final Answer:
$f^{\prime }(x)=20(5x+3{)}^3$

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Example 39: Finding the Derivative of a Function with Absolute Value

Problem:
Find the derivative of $f(x)=|3x–2|$.

Answer:
Step 1: Given Data:
$f(x)=|3x–2|$

Step 2: Solution:
The derivative is defined piecewise:
$f^{\prime }(x)=\{\begin{array}{llll}3& \text{if }3x–2>0-3& \text{if }3x–2<0\text{undefined}& \text{if }3x–2=0\end{array}$

Step 3: Final Answer:
$f^{\prime }(x)=\{\begin{array}{llll}3& \text{if }x>\frac{2}{3}-3& \text{if }x<\frac{2}{3}\text{undefined}& \text{if }x=\frac{2}{3}\end{array}$

---

Example 40: Derivative of a Polynomial Function

Problem:
Find the derivative of $f(x)=6x^5–x^4+3x^3–2x+1$.

Answer:
Step 1: Given Data:
$f(x)=6x^5–x^4+3x^3–2x+1$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=30x^4–4x^3+9x^2–2$

Step 3: Final Answer:
$f^{\prime }(x)=30x^4–4x^3+9x^2–2$

---

Example 41: Derivative of an Exponential Function with a Coefficient

Problem:
Find the derivative of $f(x)=4e^{2x}$.

Answer:
Step 1: Given Data:
$f(x)=4e^{2x}$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=4e^{2x}\cdot 2$
$=8e^{2x}$

Step 3: Final Answer:
$f^{\prime }(x)=8e^{2x}$

---

Example 42: Derivative of a Sum of Functions

Problem:
Find the derivative of $f(x)=\sin (x)+x^2+e^x$.

Answer:
Step 1: Given Data:
$f(x)=\sin (x)+x^2+e^x$

Step 2: Solution:
Using the known derivatives:
$f^{\prime }(x)=\cos (x)+2x+e^x$

Step 3: Final Answer:
$f^{\prime }(x)=\cos (x)+2x+e^x$

---

Example 43: Implicit Differentiation of an Ellipse

Problem:
Find $\frac{dy}{dx}$ for the ellipse equation $\frac{x^2}{16}+\frac{y^2}{9}=1$.

Answer:
Step 1: Given Data:
$\frac{x^2}{16}+\frac{y^2}{9}=1$

Step 2: Solution:
Differentiating both sides:
$\frac{2x}{16}+\frac{2y}{9}\frac{dy}{dx}=0$
$\frac{1}{8}x+\frac{2y}{9}\frac{dy}{dx}=0$
Thus,
$\frac{dy}{dx}=-\frac{9x}{16y}$

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

---

Example 44: Finding the Derivative of a Root Function

Problem:
Find the derivative of $f(x)=\sqrt{x^2+1}$.

Answer:
Step 1: Given Data:
$f(x)=\sqrt{x^2+1}$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{2\sqrt{x^2+1}}\cdot \frac{d}{dx}(x^2+1)$
$=\frac{1}{2\sqrt{x^2+1}}\cdot 2x$
$=\frac{x}{\sqrt{x^2+1}}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{x}{\sqrt{x^2+1}}$

---

Example 45: Differentiating a Cubic Function with Multiple Terms

Problem:
Find the derivative of $f(x)=2x^3+3x^2–5x+4$.

Answer:
Step 1: Given Data:
$f(x)=2x^3+3x^2–5x+4$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=6x^2+6x–5$

Step 3: Final Answer:
$f^{\prime }(x)=6x^2+6x–5$

---

Example 46: Derivative of a Trigonometric Function with a Coefficient

Problem:
Find the derivative of $f(x)=3\sin (4x)$.

Answer:
Step 1: Given Data:
$f(x)=3\sin (4x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=3\cdot \cos (4x)\cdot \frac{d}{dx}(4x)$
$=12\cos (4x)$

Step 3: Final Answer:
$f^{\prime }(x)=12\cos (4x)$

---

Example 47: Finding the Derivative of an Exponential Function with a Base

Problem:
Find the derivative of $f(x)={10}^x$.

Answer:
Step 1: Given Data:
$f(x)={10}^x$

Step 2: Solution:
Using the derivative of an exponential function:
$f^{\prime }(x)={10}^x\ln (10)$

Step 3: Final Answer:
$f^{\prime }(x)={10}^x\ln (10)$

---

Example 48: Derivative of a Sinusoidal Function with a Phase Shift

Problem:
Find the derivative of $f(x)=\sin (x+\frac{\pi }{4})$.

Answer:
Step 1: Given Data:
$f(x)=\sin (x+\frac{\pi }{4})$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\cos (x+\frac{\pi }{4})\cdot \frac{d}{dx}(x+\frac{\pi }{4})$
$=\cos (x+\frac{\pi }{4})$

Step 3: Final Answer:
$f^{\prime }(x)=\cos (x+\frac{\pi }{4})$

---

Example 49: Derivative of a Cotangent Function

Problem:
Find the derivative of $f(x)=\cot (x)$.

Answer:
Step 1: Given Data:
$f(x)=\cot (x)$

Step 2: Solution:
Using the known derivative:
$f^{\prime }(x)=-{\csc }^2(x)$

Step 3: Final Answer:
$f^{\prime }(x)=-{\csc }^2(x)$

---

Example 50: Derivative of a Cosecant Function

Problem:
Find the derivative of $f(x)=\csc (x)$.

Answer:
Step 1: Given Data:
$f(x)=\csc (x)$

Step 2: Solution:
Using the known derivative:
$f^{\prime }(x)=-\csc (x)\cot (x)$

Step 3: Final Answer:
$f^{\prime }(x)=-\csc (x)\cot (x)$

---

Example 51: Derivative of a Secant Function

Problem:
Find the derivative of $f(x)=\sec (x)$.

Answer:
Step 1: Given Data:
$f(x)=\sec (x)$

Step 2: Solution:
Using the known derivative:
$f^{\prime }(x)=\sec (x)\tan (x)$

Step 3: Final Answer:
$f^{\prime }(x)=\sec (x)\tan (x)$

---

Example 52: Derivative of a Composite Function with Trigonometric Functions

Problem:
Find the derivative of $f(x)=\sin (3x^2+2)$.

Answer:
Step 1: Given Data:
$f(x)=\sin (3x^2+2)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\cos (3x^2+2)\cdot \frac{d}{dx}(3x^2+2)$
$=\cos (3x^2+2)\cdot 6x$

Step 3: Final Answer:
$f^{\prime }(x)=6x\cos (3x^2+2)$

---

Example 53: Derivative of a Function with a Logarithmic Function

Problem:
Find the derivative of $f(x)=x\ln (x)$.

Answer:
Step 1: Given Data:
$f(x)=x\ln (x)$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=\frac{d}{dx}(x)\ln (x)+x\frac{d}{dx}(\ln (x))$
$=1\cdot \ln (x)+x\cdot \frac{1}{x}$
$=\ln (x)+1$

Step 3: Final Answer:
$f^{\prime }(x)=\ln (x)+1$

---

Example 54: Finding the Derivative of a Rational Function

Problem:
Find the derivative of $f(x)=\frac{1}{x}+2x^3$.

Answer:
Step 1: Given Data:
$f(x)=\frac{1}{x}+2x^3$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=-\frac{1}{x^2}+6x^2$

Step 3: Final Answer:
$f^{\prime }(x)=-\frac{1}{x^2}+6x^2$

---

Example 55: Derivative of a Function with Multiple Terms

Problem:
Find the derivative of $f(x)=5x^4+4x^2–2x+1$.

Answer:
Step 1: Given Data:
$f(x)=5x^4+4x^2–2x+1$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=20x^3+8x–2$

Step 3: Final Answer:
$f^{\prime }(x)=20x^3+8x–2$

---

Example 56: Derivative of a Function Defined Piecewise

Problem:
Find the derivative of
$f(x)=\{\begin{array}{lll}{x}^3& \text{if }x<12x& \text{if }x\ge 1\end{array}$.

Answer:
Step 1: Given Data:
$f(x)=\{\begin{array}{lll}{x}^3& \text{if }x<12x& \text{if }x\ge 1\end{array}$

Step 2: Solution:
For $x<1$:
$f^{\prime }(x)=3x^2$

For $x\ge 1$:
$f^{\prime }(x)=2$

Step 3: Final Answer:
$f^{\prime }(x)=\{\begin{array}{lll}3x^2& \text{if }x<12& \text{if }x\ge 1\end{array}$

---

Example 57: Differentiating an Exponential Function with a Shift

Problem:
Find the derivative of $f(x)=e^{x+1}$.

Answer:
Step 1: Given Data:
$f(x)=e^{x+1}$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=e^{x+1}\cdot \frac{d}{dx}(x+1)$
$=e^{x+1}\cdot 1$

Step 3: Final Answer:
$f^{\prime }(x)=e^{x+1}$

---

Example 58: Derivative of a Polynomial Function with a Higher Degree

Problem:
Find the derivative of $f(x)=2x^5–3x^4+x^3–7$.

Answer:
Step 1: Given Data:
$f(x)=2x^5–3x^4+x^3–7$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=10x^4–12x^3+3x^2$

Step 3: Final Answer:
$f^{\prime }(x)=10x^4–12x^3+3x^2$

---

Example 59: Derivative of a Function Involving a Constant

Problem:
Find the derivative of $f(x)=2x^3+7x+5$.

Answer:
Step 1: Given Data:
$f(x)=2x^3+7x+5$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=6x^2+7$

Step 3: Final Answer:
$f^{\prime }(x)=6x^2+7$

---

Example 60: Finding the Derivative of a Complex Function

Problem:
Find the derivative of $f(x)=\frac{x^2+1}{x^2–1}$.

Answer:
Step 1: Given Data:
$f(x)=\frac{x^2+1}{x^2–1}$

Step 2: Solution:
Using the quotient rule:
$f^{\prime }(x)=\frac{(2x)(x^2–1)–(x^2+1)(2x)}{(x^2–1{)}^2}$
$=\frac{2x^3–2x–2x^3–2x}{(x^2–1{)}^2}$
$=\frac{-4x}{(x^2–1{)}^2}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{-4x}{(x^2–1{)}^2}$

---

Example 61: Derivative of a Logarithmic Function with a Coefficient

Problem:
Find the derivative of $f(x)=3\ln (2x+1)$.

Answer:
Step 1: Given Data:
$f(x)=3\ln (2x+1)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=3\cdot \frac{1}{2x+1}\cdot 2$
$=\frac{6}{2x+1}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{6}{2x+1}$

---

Example 62: Finding the Derivative of a Function with an Absolute Value

Problem:
Find the derivative of $f(x)=|x^2–4|$.

Answer:
Step 1: Given Data:
$f(x)=|x^2–4|$

Step 2: Solution:
The derivative is defined piecewise:
$f^{\prime }(x)=\{\begin{array}{llll}2x& \text{if }{x}^2–4>0-2x& \text{if }{x}^2–4<0\text{undefined}& \text{if }{x}^2–4=0\end{array}$

Step 3: Final Answer:
$f^{\prime }(x)=\{\begin{array}{llll}2x& \text{if }|x|>2-2x& \text{if }|x|<2\text{undefined}& \text{if }|x|=2\end{array}$

---

Example 63: Derivative of a Sinusoidal Function with a Phase Shift

Problem:
Find the derivative of $f(x)=\sin (2x–\frac{\pi }{3})$.

Answer:
Step 1: Given Data:
$f(x)=\sin (2x–\frac{\pi }{3})$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\cos (2x–\frac{\pi }{3})\cdot \frac{d}{dx}(2x–\frac{\pi }{3})$
$=2\cos (2x–\frac{\pi }{3})$

Step 3: Final Answer:
$f^{\prime }(x)=2\cos (2x–\frac{\pi }{3})$

---

Example 64: Derivative of a Function with a Trigonometric and Polynomial Component

Problem:
Find the derivative of $f(x)=x^2\sin (x)+x$.

Answer:
Step 1: Given Data:
$f(x)=x^2\sin (x)+x$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=\frac{d}{dx}(x^2)\sin (x)+x^2\frac{d}{dx}(\sin (x))+\frac{d}{dx}(x)$
$=2x\sin (x)+x^2\cos (x)+1$

Step 3: Final Answer:
$f^{\prime }(x)=2x\sin (x)+x^2\cos (x)+1$

---

Example 65: Finding the Derivative of a Natural Logarithm Function

Problem:
Find the derivative of $f(x)=\ln (5x^3)$.

Answer:
Step 1: Given Data:
$f(x)=\ln (5x^3)$

Step 2: Solution:
Using the properties of logarithms:
$f(x)=\ln (5)+3\ln (x)$
Thus,
$f^{\prime }(x)=0+\frac{3}{x}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{3}{x}$

---

Example 66: Derivative of an Arc Function

Problem:
Find the derivative of $f(x)=\arcsin (x)+\arccos (x)$.

Answer:
Step 1: Given Data:
$f(x)=\arcsin (x)+\arccos (x)$

Step 2: Solution:
Using the known derivatives:
$f^{\prime }(x)=\frac{1}{\sqrt{1–x^2}}–\frac{1}{\sqrt{1–x^2}}=0$

Step 3: Final Answer:
$f^{\prime }(x)=0$

---

Example 67: Finding the Derivative of a Complex Function

Problem:
Find the derivative of $f(x)=e^{x^2}\cos (x)$.

Answer:
Step 1: Given Data:
$f(x)=e^{x^2}\cos (x)$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=e^{x^2}\cdot \frac{d}{dx}(\cos (x))+\cos (x)\cdot \frac{d}{dx}(e^{x^2})$
$=e^{x^2}\cdot (-\sin (x))+\cos (x)\cdot (2x{e}^{x^2})$
$=-e^{x^2}\sin (x)+2x{e}^{x^2}\cos (x)$

Step 3: Final Answer:
$f^{\prime }(x)=-e^{x^2}\sin (x)+2x{e}^{x^2}\cos (x)$

---

Example 68: Derivative of an Absolute Value Function

Problem:
Find the derivative of $f(x)=|x^3–1|$.

Answer:
Step 1: Given Data:
$f(x)=|x^3–1|$

Step 2: Solution:
The derivative is defined piecewise:
$f^{\prime }(x)=\{\begin{array}{llll}3x^2& \text{if }{x}^3–1>0-3x^2& \text{if }{x}^3–1<0\text{undefined}& \text{if }{x}^3–1=0\end{array}$

Step 3: Final Answer:
$f^{\prime }(x)=\{\begin{array}{llll}3x^2& \text{if }x>1-3x^2& \text{if }x<1\text{undefined}& \text{if }x=1\end{array}$

---

Example 69: Derivative of a Polynomial Function with Multiple Variables

Problem:
Find the derivative of $f(x)=2x^{3}y+3x{y}^2$ with respect to $x$.

Answer:
Step 1: Given Data:
$f(x)=2x^{3}y+3x{y}^2$

Step 2: Solution:
Using the product rule for each term:
$\frac{df}{dx}=6x^{2}y+3y^2$

Step 3: Final Answer:
$\frac{df}{dx}=6x^{2}y+3y^2$

---

Example 70: Derivative of a Function with an Implicit Equation

Problem:
Find $\frac{dy}{dx}$ for the equation $xy+y^2=7$.

Answer:
Step 1: Given Data:
$xy+y^2=7$

Step 2: Solution:
Differentiating both sides:
$y+x\frac{dy}{dx}+2y\frac{dy}{dx}=0$
Thus,
$(x+2y)\frac{dy}{dx}=-y$
Therefore,
$\frac{dy}{dx}=-\frac{y}{x+2y}$

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

---

Example 71: Finding the Derivative of a Cubic Function

Problem:
Find the derivative of $f(x)=2x^3–3x^2+5$.

Answer:
Step 1: Given Data:
$f(x)=2x^3–3x^2+5$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=6x^2–6x$

Step 3: Final Answer:
$f^{\prime }(x)=6x^2–6x$

---

Example 72: Derivative of a Composite Function with a Logarithm

Problem:
Find the derivative of $f(x)=\ln (x^2+3x)$.

Answer:
Step 1: Given Data:
$f(x)=\ln (x^2+3x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{x^2+3x}\cdot (2x+3)$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{2x+3}{x^2+3x}$

---

Example 73: Derivative of a Function with a Product of Functions

Problem:
Find the derivative of $f(x)=(x^2+1)(\sin (x))$.

Answer:
Step 1: Given Data:
$f(x)=(x^2+1)(\sin (x))$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=(x^2+1)\cos (x)+\sin (x)(2x)$
$=(x^2+1)\cos (x)+2x\sin (x)$

Step 3: Final Answer:
$f^{\prime }(x)=(x^2+1)\cos (x)+2x\sin (x)$

---

Example 74: Derivative of a Cubic Function with a Constant

Problem:
Find the derivative of $f(x)=5x^3+3$.

Answer:
Step 1: Given Data:
$f(x)=5x^3+3$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=15x^2$

Step 3: Final Answer:
$f^{\prime }(x)=15x^2$

---

Example 75: Finding the Derivative of a Function with a Sum

Problem:
Find the derivative of $f(x)=x^2+\frac{1}{x}+\ln (x)$.

Answer:
Step 1: Given Data:
$f(x)=x^2+\frac{1}{x}+\ln (x)$

Step 2: Solution:
Using the known derivatives:
$f^{\prime }(x)=2x–\frac{1}{x^2}+\frac{1}{x}$
$=2x–\frac{1}{x^2}+\frac{1}{x}$

Step 3: Final Answer:
$f^{\prime }(x)=2x–\frac{1}{x^2}+\frac{1}{x}$

---

Example 76: Derivative of an Inverse Trigonometric Function

Problem:
Find the derivative of $f(x)=\arcsin (2x)$.

Answer:
Step 1: Given Data:
$f(x)=\arcsin (2x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{\sqrt{1–(2x{)}^2}}\cdot \frac{d}{dx}(2x)$
$=\frac{2}{\sqrt{1–4x^2}}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{2}{\sqrt{1–4x^2}}$

---

Example 77: Derivative of a Function with an Exponential and Polynomial Component

Problem:
Find the derivative of $f(x)=x^3{e}^x$.

Answer:
Step 1: Given Data:
$f(x)=x^3{e}^x$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=\frac{d}{dx}(x^3)e^x+x^3\frac{d}{dx}(e^x)$
$=3x^2{e}^x+x^3{e}^x$
$=e^x(3x^2+x^3)$

Step 3: Final Answer:
$f^{\prime }(x)=e^x(3x^2+x^3)$

---

Example 78: Derivative of a Function with Multiple Variables

Problem:
Find the derivative of $f(x,y)=x^{2}y+x{y}^2$ with respect to $x$.

Answer:
Step 1: Given Data:
$f(x,y)=x^{2}y+x{y}^2$

Step 2: Solution:
Using the product rule for each term:
$\frac{df}{dx}=2xy+y^2$

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

---

Example 79: Finding the Derivative of a Logarithmic Function with a Composite Argument

Problem:
Find the derivative of $f(x)=\ln (3x^2+2x)$.

Answer:
Step 1: Given Data:
$f(x)=\ln (3x^2+2x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{3x^2+2x}\cdot (6x+2)$
$=\frac{6x+2}{3x^2+2x}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{6x+2}{3x^2+2x}$

---

Example 80: Derivative of a Quadratic Function

Problem:
Find the derivative of $f(x)=x^2+2x+1$.

Answer:
Step 1: Given Data:
$f(x)=x^2+2x+1$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=2x+2$

Step 3: Final Answer:
$f^{\prime }(x)=2x+2$

---

Example 81: Derivative of a Function with a Higher Degree

Problem:
Find the derivative of $f(x)=5x^6–3x^4+2x^2$.

Answer:
Step 1: Given Data:
$f(x)=5x^6–3x^4+2x^2$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=30x^5–12x^3+4x$

Step 3: Final Answer:
$f^{\prime }(x)=30x^5–12x^3+4x$

---

Example 82: Derivative of a Function with an Arc Function

Problem:
Find the derivative of $f(x)=\arccos (x)+\arcsin (x)$.

Answer:
Step 1: Given Data:
$f(x)=\arccos (x)+\arcsin (x)$

Step 2: Solution:
Using the known derivatives:
$f^{\prime }(x)=-\frac{1}{\sqrt{1–x^2}}+\frac{1}{\sqrt{1–x^2}}=0$

Step 3: Final Answer:
$f^{\prime }(x)=0$

---

Example 83: Derivative of a Product Function

Problem:
Find the derivative of $f(x)=(2x+1)(x^2–4)$.

Answer:
Step 1: Given Data:
$f(x)=(2x+1)(x^2–4)$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=(2)(x^2–4)+(2x+1)(2x)$
$=2x^2–8+4x^2+2x$
$=6x^2+2x–8$

Step 3: Final Answer:
$f^{\prime }(x)=6x^2+2x–8$

---

Example 84: Derivative of a Function with Nested Functions

Problem:
Find the derivative of $f(x)=\sin (\ln (x))$.

Answer:
Step 1: Given Data:
$f(x)=\sin (\ln (x))$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\cos (\ln (x))\cdot \frac{d}{dx}(\ln (x))$
$=\cos (\ln (x))\cdot \frac{1}{x}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{\cos (\ln (x))}{x}$

---

Example 85: Derivative of a Function with a Combination of Functions

Problem:
Find the derivative of $f(x)=e^{x^2}\cos (x)+x^3$.

Answer:
Step 1: Given Data:
$f(x)=e^{x^2}\cos (x)+x^3$

Step 2: Solution:
Using the product rule on the first term:
$f^{\prime }(x)=\frac{d}{dx}(e^{x^2})\cos (x)+e^{x^2}\frac{d}{dx}(\cos (x))+3x^2$
$=2x{e}^{x^2}\cos (x)–e^{x^2}\sin (x)+3x^2$

Step 3: Final Answer:
$f^{\prime }(x)=2x{e}^{x^2}\cos (x)–e^{x^2}\sin (x)+3x^2$

---

Example 86: Derivative of a Polynomial with Multiple Terms

Problem:
Find the derivative of $f(x)=7x^4–2x^2+x–3$.

Answer:
Step 1: Given Data:
$f(x)=7x^4–2x^2+x–3$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=28x^3–4x+1$

Step 3: Final Answer:
$f^{\prime }(x)=28x^3–4x+1$

---

Example 87: Derivative of a Logarithmic Function with a Rational Argument

Problem:
Find the derivative of $f(x)=\ln \left(\frac{x^2+1}{x^2}\right)$.

Answer:
Step 1: Given Data:
$f(x)=\ln \left(\frac{x^2+1}{x^2}\right)$

Step 2: Solution:
Using the properties of logarithms:
$f(x)=\ln (x^2+1)–\ln (x^2)$
$=\ln (x^2+1)–2\ln (x)$

Now differentiate:
$f^{\prime }(x)=\frac{2x}{x^2+1}–\frac{2}{x}$
$=\frac{2x^2–2(x^2+1)}{x(x^2+1)}$
$=\frac{-2}{x(x^2+1)}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{-2}{x(x^2+1)}$

---

Example 88: Derivative of an Arc Function with a Coefficient

Problem:
Find the derivative of $f(x)=4\arcsin (2x)$.

Answer:
Step 1: Given Data:
$f(x)=4\arcsin (2x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=4\cdot \frac{1}{\sqrt{1–(2x{)}^2}}\cdot \frac{d}{dx}(2x)$
$=\frac{8}{\sqrt{1–4x^2}}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{8}{\sqrt{1–4x^2}}$

---

Example 89: Derivative of a Sinusoidal Function with Multiple Angles

Problem:
Find the derivative of $f(x)=\sin (3x)+\cos (2x)$.

Answer:
Step 1: Given Data:
$f(x)=\sin (3x)+\cos (2x)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=3\cos (3x)–2\sin (2x)$

Step 3: Final Answer:
$f^{\prime }(x)=3\cos (3x)–2\sin (2x)$

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Example 90: Finding the Derivative of a Function with a Rational Component

Problem:
Find the derivative of $f(x)=\frac{1}{x^3+1}$.

Answer:
Step 1: Given Data:
$f(x)=\frac{1}{x^3+1}$

Step 2: Solution:
Using the quotient rule:
$f^{\prime }(x)=\frac{(0)(x^3+1)–(1)(3x^2)}{(x^3+1{)}^2}$
$=\frac{-3x^2}{(x^3+1{)}^2}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{-3x^2}{(x^3+1{)}^2}$

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Example 91: Derivative of a Composite Function with a Polynomial and Trigonometric Component

Problem:
Find the derivative of $f(x)=x^2\sin (x^3)$.

Answer:
Step 1: Given Data:
$f(x)=x^2\sin (x^3)$

Step 2: Solution:
Using the product rule:
$f^{\prime }(x)=2x\sin (x^3)+x^2\cdot \cos (x^3)\cdot 3x^2$
$=2x\sin (x^3)+3x^4\cos (x^3)$

Step 3: Final Answer:
$f^{\prime }(x)=2x\sin (x^3)+3x^4\cos (x^3)$

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Example 92: Finding the Derivative of a Function with Multiple Variables

Problem:
Find the derivative of $f(x,y)=x{y}^2+3y$ with respect to $y$.

Answer:
Step 1: Given Data:
$f(x,y)=x{y}^2+3y$

Step 2: Solution:
Using the product rule:
$\frac{df}{dy}=x\cdot 2y+3$
$=2xy+3$

Step 3: Final Answer:
$\frac{df}{dy}=2xy+3$

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Example 93: Derivative of a Trigonometric Function with a Shift

Problem:
Find the derivative of $f(x)=\tan (2x+1)$.

Answer:
Step 1: Given Data:
$f(x)=\tan (2x+1)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)={\sec }^2(2x+1)\cdot \frac{d}{dx}(2x+1)$
$=2{\sec }^2(2x+1)$

Step 3: Final Answer:
$f^{\prime }(x)=2{\sec }^2(2x+1)$

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Example 94: Finding the Derivative of a Complex Function

Problem:
Find the derivative of $f(x)=\frac{x^2+3}{\sin (x)}$.

Answer:
Step 1: Given Data:
$f(x)=\frac{x^2+3}{\sin (x)}$

Step 2: Solution:
Using the quotient rule:
$f^{\prime }(x)=\frac{(\sin (x))(2x)–(x^2+3)(\cos (x))}{{\sin }^2(x)}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{2x\sin (x)–(x^2+3)\cos (x)}{{\sin }^2(x)}$

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Example 95: Derivative of an Inverse Function

Problem:
Find the derivative of $f(x)=\arctan (3x+1)$.

Answer:
Step 1: Given Data:
$f(x)=\arctan (3x+1)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{1+(3x+1{)}^2}\cdot \frac{d}{dx}(3x+1)$
$=\frac{3}{1+(3x+1{)}^2}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{3}{1+(3x+1{)}^2}$

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Example 96: Finding the Derivative of a Function with an Arc Function

Problem:
Find the derivative of $f(x)=\arcsin (x^2)$.

Answer:
Step 1: Given Data:
$f(x)=\arcsin (x^2)$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{\sqrt{1–(x^2{)}^2}}\cdot \frac{d}{dx}(x^2)$
$=\frac{1}{\sqrt{1–x^4}}\cdot 2x$
$=\frac{2x}{\sqrt{1–x^4}}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{2x}{\sqrt{1–x^4}}$

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Example 97: Derivative of a Function with Nested Logarithms

Problem:
Find the derivative of $f(x)=\ln (\ln (x))$.

Answer:
Step 1: Given Data:
$f(x)=\ln (\ln (x))$

Step 2: Solution:
Using the chain rule:
$f^{\prime }(x)=\frac{1}{\ln (x)}\cdot \frac{1}{x}$
$=\frac{1}{x\ln (x)}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{1}{x\ln (x)}$

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Example 98: Derivative of a Polynomial Function with Mixed Terms

Problem:
Find the derivative of $f(x)=x^4–2x^3+x^2–5x+2$.

Answer:
Step 1: Given Data:
$f(x)=x^4–2x^3+x^2–5x+2$

Step 2: Solution:
Using the power rule:
$f^{\prime }(x)=4x^3–6x^2+2x–5$

Step 3: Final Answer:
$f^{\prime }(x)=4x^3–6x^2+2x–5$

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Example 99: Finding the Derivative of a Function with Multiple Variables

Problem:
Find the derivative of $f(x,y)=x^3+3xy+y^2$ with respect to $y$.

Answer:
Step 1: Given Data:
$f(x,y)=x^3+3xy+y^2$

Step 2: Solution:
Using the product rule:
$\frac{df}{dy}=3x+2y$

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

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Example 100: Derivative of a Function with an Arc Function

Problem:
Find the derivative of $f(x)=5\arctan (x)$.

Answer:
Step 1: Given Data:
$f(x)=5\arctan (x)$

Step 2: Solution:
Using the known derivative:
$f^{\prime }(x)=5\cdot \frac{1}{1+x^2}$

Step 3: Final Answer:
$f^{\prime }(x)=\frac{5}{1+x^2}$