The Chain Rule and Composite Functions-Chapter 3

Additional Derivative Topics

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 formed by combining two or more functions, which is essential in business-calculus applications where complex relationships often arise.

Composite Functions

A composite function is formed when one function is applied to the result of another function. If f and g are functions, the composite function m is defined as:

• Definition: $m(x) = f(g(x))$

• Domain: The domain of m is all values of x such that x is in the domain of g and g(x) is in the domain of f.

Example: If $f(x) = x^2$ and $g(x) = x^3$, then $m(x) = f(g(x)) = (x^3)^2 = x^6$.

Writing Functions as Composites

Many functions can be expressed as composites of simpler functions. This is useful for applying the chain rule.

• Example: Let $h(x) = \sqrt{1 + x^2}$. We can write $h(x)$ as $f(g(x))$ where $f(x) = \sqrt{x}$ and $g(x) = 1 + x^2$.

The Power Rule for Composite Functions

The power rule is a basic derivative rule, but it can be extended to composite functions using the chain rule.

• Standard Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• General Power Rule: If $u(x)$ is a differentiable function and $n$ is any real number, then:

$\frac{d}{dx} [u(x)]^n = n [u(x)]^{n-1} \cdot u'(x)$

Example: Find the derivative of $[3x + 2]^4$:

• Let $u(x) = 3x + 2$, $n = 4$

• $\frac{d}{dx} [3x + 2]^4 = 4[3x + 2]^3 \cdot 3 = 12[3x + 2]^3$

The Chain Rule

The chain rule provides a method for differentiating composite functions. It states:

• Theorem: If $m(x) = f(g(x))$, then $m'(x) = f'(g(x)) \cdot g'(x)$, provided $f'$ and $g'$ exist.

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

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

Applying the Chain Rule

To differentiate a composite function:

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

2. Find the derivative of the outer function evaluated at the inner function: $f'(g(x))$.

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

Example: Find the derivative of $y = (2x + 5)^3$:

• Outer function: $f(x) = x^3$

• Inner function: $g(x) = 2x + 5$

• $y' = 3(2x + 5)^2 \cdot 2 = 6(2x + 5)^2$

General Derivative Rules

The chain rule generalizes basic derivative rules, allowing for differentiation of more complex functions.

• Example: Find the derivative of $y = \ln(5x^2 + 1)$:

• Outer function: $f(x) = \ln(x)$

• Inner function: $g(x) = 5x^2 + 1$

• $y' = \frac{1}{5x^2 + 1} \cdot 10x = \frac{10x}{5x^2 + 1}$

Summary Table: Chain Rule Applications

Function	Outer Function	Inner Function	Derivative
$[3x + 2]^4$	$x^4$	$3x + 2$	$12[3x + 2]^3$
$\sin(x^2)$	$\sin(x)$	$x^2$	$\cos(x^2) \cdot 2x$
$\ln(5x^2 + 1)$	$\ln(x)$	$5x^2 + 1$	$\frac{10x}{5x^2 + 1}$