The Chain Rule; Derivatives of Exponential, Logarithmic & Inverse Trigonometric Functions

The Chain Rule

Introduction

The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. It allows us to differentiate functions that are formed by the composition of two or more functions, and is essential for handling complex expressions in calculus.

The Chain Rule - Leibniz Notation

Given a composite function $y = f(g(x))$, where $u = g(x)$ and $y = f(u)$, the derivative of $y$ with respect to $x$ is:

• Formula: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

• Steps to Apply the Chain Rule:

  1. Identify the outer function $f$ and the inner function $g$.

  2. Let $u = g(x)$ and $y = f(u)$.

  3. Find the product $\frac{dy}{du} \cdot \frac{du}{dx}$.

  4. Express the result in terms of $x$.

• Example: If $y = \cos(x^2)$, let $u = x^2$, $y = \cos(u)$. $\frac{dy}{dx} = -\sin(u) \cdot 2x = -2x \sin(x^2)$

The Chain Rule - Prime Notation

Prime notation is another way to express the chain rule, especially useful for compositions:

• Formula: $y' = f'(g(x)) \cdot g'(x)$

• Steps:

  1. Identify the outer function $f$ and the inner function $g$.

  2. Take the derivative of the outer function, $f'$, and evaluate at the inner function: $f'(g(x))$.

  3. Multiply by the derivative of the inner function: $g'(x)$.

• Example: If $y = \sin(x^2)$, then $f(x) = \sin x$, $g(x) = x^2$. $f'(x) = \cos x$, $g'(x) = 2x$ $y' = \cos(x^2) \cdot 2x$

Composition of Three or More Functions

When differentiating a function composed of three or more nested functions, the chain rule is applied repeatedly:

• Formula: $\frac{dy}{dx} = \frac{df}{dh} \cdot \frac{dh}{dg} \cdot \frac{dg}{dx}$

• Example: For $y = \sin(3x^2 + 5)$: Let $u = 3x^2 + 5$, $y = \sin(u)$ $\frac{dy}{dx} = \cos(u) \cdot 6x = 6x \cos(3x^2 + 5)$

Derivatives of Exponential Functions

Introduction

Exponential functions are of the form $y = a^{x}$ or $y = e^{x}$. Their derivatives have unique properties that make them important in calculus and applications.

• Key Formulas:

  • $\frac{d}{dx} [e^{x}] = e^{x}$

  • $\frac{d}{dx} [a^{x}] = a^{x} \ln a$

• Example: $y = e^{\cot x}$ $\frac{dy}{dx} = e^{\cot x} \cdot (-\csc^2 x)$

Derivatives of Logarithmic Functions

Introduction

Logarithmic functions are the inverses of exponential functions. Their derivatives are useful in many calculus problems, especially those involving growth and decay.

• Key Formulas:

  • $\frac{d}{dx} [\ln x] = \frac{1}{x}$

  • $\frac{d}{dx} [\log_a x] = \frac{1}{x \ln a}$

• Example: $y = 3x^2 \log_3 x$ Use product and chain rule as needed.

Derivatives of Inverse Trigonometric Functions

Introduction

Inverse trigonometric functions, such as $\arctan x$, $\arccos x$, and $\arcsec x$, have specific derivative formulas. These are important for solving integrals and differential equations involving trigonometric expressions.

• Key Formulas:

  • $\frac{d}{dx} [\tan^{-1} x] = \frac{1}{1 + x^2}$, for $-\infty < x < \infty$

  • $\frac{d}{dx} [\sec^{-1} x] = \frac{1}{|x| \sqrt{x^2 - 1}}$, for $|x| > 1$

  • $\frac{d}{dx} [\cot^{-1} x] = \frac{-1}{1 + x^2}$, for $-\infty < x < \infty$

  • $\frac{d}{dx} [\csc^{-1} x] = \frac{-1}{|x| \sqrt{x^2 - 1}}$, for $|x| > 1$

• Example: $y = \sec^{-1}(2x)$ $\frac{dy}{dx} = \frac{1}{|2x| \sqrt{(2x)^2 - 1}} \cdot 2 = \frac{2}{|2x| \sqrt{4x^2 - 1}}$

Summary Table: Derivative Formulas

Function	Derivative	Domain
$e^{x}$	$e^{x}$	All real $x$
$a^{x}$	$a^{x} \ln a$	All real $x$
$\ln x$	$\frac{1}{x}$	$x > 0$
$\log_a x$	$\frac{1}{x \ln a}$	$x > 0$
$\tan^{-1} x$	$\frac{1}{1 + x^2}$	All real $x$
$\sec^{-1} x$	$\frac{1}{|x| \sqrt{x^2 - 1}}$	$|x| > 1$
$\cot^{-1} x$	$\frac{-1}{1 + x^2}$	All real $x$
$\csc^{-1} x$	$\frac{-1}{|x| \sqrt{x^2 - 1}}$	$|x| > 1$

Additional Examples and Applications

• Example: Find the derivative of $y = (4 + 5x)^6$ $\frac{dy}{dx} = 6(4 + 5x)^5 \cdot 5 = 30(4 + 5x)^5$

• Example: Find the derivative of $y = \tan x \sqrt{x}$ Use product and chain rule: $\frac{dy}{dx} = \sec^2 x \sqrt{x} + \tan x \cdot \frac{1}{2\sqrt{x}}$

• Example: Find the derivative of $y = \frac{1 + x^2}{\cot^{-1} x}$ Use quotient rule and chain rule as needed.

Additional info: These notes cover the essential rules and examples for differentiating composite, exponential, logarithmic, and inverse trigonometric functions, as required in a standard Calculus I or II college course.