The Chain Rule: Derivatives of Composite Functions

Section 3.7: The Chain Rule

Introduction to the Chain Rule

The Chain Rule is a fundamental technique in calculus for finding the derivative of composite functions. It allows us to differentiate functions that are composed of two or more functions, such as f(g(x)). The rule is essential for understanding how changes in one variable propagate through nested functions.

• Composite Function: A function formed by applying one function to the result of another, e.g., f(g(x)).

• Key Idea: The rate of change of the outer function depends on the rate of change of the inner function.

• Applications: Used extensively in physics, engineering, and economics to model systems with multiple dependent variables.

Theorem: The Chain Rule

If y = f(u) is differentiable at u = g(x) and u = g(x) is differentiable at x, then the composite function y = f(g(x)) is differentiable at x, and its derivative can be expressed in two equivalent ways:

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

• $\frac{d}{dx} \left( f(g(x)) \right) = f'(g(x)) \cdot g'(x)$

Procedure: Using the Chain Rule

To apply the Chain Rule to a differentiable function y = f(g(x)), follow these steps:

1. Identify the outer function f and the inner function g, and let u = g(x).

2. Express g(x) with u to write y in terms of u: y = f(u).

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

4. Replace u with g(x) in $\frac{dy}{du}$ to obtain $\frac{dy}{dx}$.

Examples of the Chain Rule

Example: Composition of Three Functions

Calculate the derivative of sin(ecos x). This example demonstrates the use of the Chain Rule for three nested functions.

• Inner function: cos x

• Middle function: ecos x

• Outer function: sin(ecos x)

• Apply the Chain Rule twice:

• $\frac{d}{dx} \left( \sin(e^{\cos x}) \right) = \cos(e^{\cos x}) \cdot \frac{d}{dx}(e^{\cos x})$

• $= \cos(e^{\cos x}) \cdot e^{\cos x} \cdot \frac{d}{dx}(\cos x)$

• $= \cos(e^{\cos x}) \cdot e^{\cos x} \cdot (-\sin x)$

• $= -\sin x \cdot e^{\cos x} \cdot \cos(e^{\cos x})$

Example: Combining Rules

Find $\frac{d}{dx} \left( x^2 \sqrt{x^2 + 1} \right)$. This example combines the Product Rule and the Chain Rule.

• Product Rule: $\frac{d}{dx}(u \cdot v) = u'v + uv'$

• Chain Rule for $\sqrt{x^2 + 1}$:

• $\frac{d}{dx}(x^2 \sqrt{x^2 + 1}) = \frac{d}{dx}(x^2) \cdot \sqrt{x^2 + 1} + x^2 \cdot \frac{d}{dx}(\sqrt{x^2 + 1})$

• $= 2x \sqrt{x^2 + 1} + x^2 \cdot \frac{1}{2}(x^2 + 1)^{-1/2} \cdot 2x$

• $= 2x \sqrt{x^2 + 1} + \frac{2x^3}{2 \sqrt{x^2 + 1}}$

• $= \frac{3x^3 + 2x}{\sqrt{x^2 + 1}}$

Example: Identifying Inner and Outer Functions

For each composite function, find the inner function u = g(x) and the outer function y = f(u). Use the Chain Rule to find $\frac{dy}{dx}$.

• a. $y = (5x + 4)^{1/2}$

• b. $y = \sin x^2$

• c. $y = \sin x^2$ (interpreted as $\sin(x^2)$)

Solution: Applying the Chain Rule

• a. Inner function: $u = 5x + 4$, Outer function: $y = u^{1/2}$

• $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{1}{2}u^{-1/2} \cdot 5 = \frac{5}{2}(5x + 4)^{-1/2}$

• b. Inner function: $u = x^2$, Outer function: $y = \sin u$

• $\frac{dy}{dx} = \cos u \cdot 2x = 2x \cos x^2$

• c. Inner function: $u = x^2$, Outer function: $y = \sin u$

• $\frac{dy}{dx} = \cos u \cdot 2x = 2x \cos x^2$

Chain Rule for Powers

Theorem: Chain Rule for Powers

If g is differentiable for all x in its domain and p is a real number, then:

• $\frac{d}{dx} \left( (g(x))^p \right) = p (g(x))^{p-1} \cdot g'(x)$

Example: Chain Rule for Powers

Find $\frac{d}{dx} \left( (\tan x + 10)^{21} \right)$.

• Let $g(x) = \tan x + 10$

• Apply the Chain Rule:

• $\frac{d}{dx} \left( (\tan x + 10)^{21} \right) = 21 (\tan x + 10)^{20} \cdot \frac{d}{dx}(\tan x + 10)$

• $= 21 (\tan x + 10)^{20} \cdot \sec^2 x$

Summary Table: Chain Rule Applications

Function	Inner Function	Outer Function	Derivative
$\sin(e^{\cos x})$	$\cos x$	$\sin(e^u)$	$-\sin x \cdot e^{\cos x} \cdot \cos(e^{\cos x})$
$x^2 \sqrt{x^2 + 1}$	$x^2 + 1$	$x^2 \cdot u^{1/2}$	$\frac{3x^3 + 2x}{\sqrt{x^2 + 1}}$
$(5x + 4)^{1/2}$	$5x + 4$	$u^{1/2}$	$\frac{5}{2}(5x + 4)^{-1/2}$
$\sin x^2$	$x^2$	$\sin u$	$2x \cos x^2$
$(\tan x + 10)^{21}$	$\tan x + 10$	$u^{21}$	$21 (\tan x + 10)^{20} \sec^2 x$