Differentiation and Applications: Practice Problems and Solutions

Problems on Differentiation and Applications

Introduction

This set of problems focuses on the differentiation of various types of functions, including polynomials, rational functions, exponentials, logarithms, and trigonometric functions. It also includes applied problems involving rates of change and the use of derivatives to find tangent lines and solve composite function problems.

Differentiation of Functions

Basic Differentiation Rules

• Power Rule: If $f(x) = x^n$, then $f'(x) = n x^{n-1}$.

• Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.

• Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.

• Chain Rule: If $f(x) = h(g(x))$, then $f'(x) = h'(g(x)) \cdot g'(x)$.

Examples of Differentiation

• Polynomial Function: $f(x) = (5x^2 + 2x^4 - 3x + 1)^3$ Solution: Use the chain rule and power rule. $f'(x) = 3(5x^2 + 2x^4 - 3x + 1)^2 \cdot (10x + 8x^3 - 3)$

• Rational Function: $g(x) = \frac{x^2 - 5}{2x + 1}$ Solution: Use the quotient rule. $g'(x) = \frac{2x(2x + 1) - (x^2 - 5) \cdot 2}{(2x + 1)^2}$

• Exponential Function: $f(x) = e^{x^2}$ Solution: Use the chain rule. $f'(x) = 2x e^{x^2}$

• Logarithmic Function: $h(x) = \ln(4x^3 + 5x^2)$ Solution: Use the chain rule. $h'(x) = \frac{12x^2 + 10x}{4x^3 + 5x^2}$

• Trigonometric Function: $f(x) = 2\tan(5x^2)$ Solution: Use the chain rule. $f'(x) = 20x \sec^2(5x^2)$

Special Derivatives

• Derivative of $\sin(x)$: $\frac{d}{dx} \sin(x) = \cos(x)$

• Derivative of $\cos(x)$: $\frac{d}{dx} \cos(x) = -\sin(x)$

• Derivative of $\ln(x)$: $\frac{d}{dx} \ln(x) = \frac{1}{x}$

• Derivative of $e^{x}$: $\frac{d}{dx} e^{x} = e^{x}$

Applications of Derivatives

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another, often time. These problems require implicit differentiation and the chain rule.

• Example: If $A(t) = 25 + 3t^2$ represents the amount of a drug in the body at time $t$, then the rate of change is $A'(t) = 6t$.

Equation of the Tangent Line

The tangent line to the curve $y = f(x)$ at $x = a$ has the equation:

• $y = f(a) + f'(a)(x - a)$

To find the tangent line, compute $f(a)$ and $f'(a)$, then substitute into the formula above.

Composite Functions and Their Derivatives

When dealing with composite functions, use the chain rule and product/quotient rules as needed.

• Example: If $h(x) = e^{x^2} g(x)$, then $h'(x) = e^{x^2} g'(x) + 2x e^{x^2} g(x)$

• Example: If $h(x) = \frac{f(x)}{4x + g(x)}$, then use the quotient rule.

Summary Table: Common Derivative Rules

Function	Derivative
$x^n$	$n x^{n-1}$
$e^{x}$	$e^{x}$
$\ln(x)$	$\frac{1}{x}$
$\sin(x)$	$\cos(x)$
$\cos(x)$	$-\sin(x)$
$\tan(x)$	$\sec^2(x)$

Practice Problems and Solutions

• Differentiation: Practice differentiating a variety of functions using the rules above.

• Applications: Apply derivatives to solve problems involving rates of change, tangent lines, and composite functions.

Additional info: The problems and solutions provided are typical of early Calculus courses, focusing on differentiation techniques and their applications. The table above summarizes the most common derivative rules for reference.