The Chain Rule and Differentiation of Composite Functions: 3.7

3.7: 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 combining two or more basic functions. This section reviews the derivatives of basic functions and demonstrates how to apply the chain rule to various types of composite functions.

Basic Function Derivatives

Common Derivative Formulas

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

• Exponential Function: For $y = e^x$, $\frac{d}{dx}e^x = e^x$

• Trigonometric Functions:

  • $\frac{d}{dx}\sin x = \cos x$

  • $\frac{d}{dx}\cos x = -\sin x$

  • $\frac{d}{dx}\tan x = \sec^2 x$

  • $\frac{d}{dx}\cot x = -\csc^2 x$

  • $\frac{d}{dx}\sec x = \sec x \tan x$

  • $\frac{d}{dx}\csc x = -\csc x \cot x$

Composite functions are formed by combining these basic functions, such as $\sqrt{\sin x}$, $\tan(e^x + x^2)$, or $e^{x \cos x}$.

The Chain Rule

Statement of the Chain Rule

The chain rule provides a method for differentiating composite functions. If $y = f(u)$ and $u = g(x)$, then:

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

• In function notation: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$

Key Point: The derivative of the outer function is evaluated at the inner function, then multiplied by the derivative of the inner function.

Examples of the Chain Rule

Example 1: $y = \sin(12x - 7)$

• Let $u = 12x - 7$

• $\frac{d}{dx}\sin(u) = \cos(u) \cdot u'$

• $y' = \cos(12x - 7) \cdot 12 = 12\cos(12x - 7)$

Example 2: $y = \tan(e^x)$

• Let $u = e^x$

• $\frac{d}{dx}\tan(u) = \sec^2(u) \cdot u'$

• $y' = \sec^2(e^x) \cdot e^x = e^x \sec^2(e^x)$

Example 3: $y = e^{3x^2 + 1}$

• Let $u = 3x^2 + 1$

• $\frac{d}{dx}e^u = e^u \cdot u'$

• $y' = e^{3x^2 + 1} \cdot 6x = 6x e^{3x^2 + 1}$

Example 4: $y = (3x^2 + 2)^6$

• Let $u = 3x^2 + 2$

• $\frac{d}{dx}u^6 = 6u^5 \cdot u'$

• $y' = 6(3x^2 + 2)^5 \cdot 6x = 36x(3x^2 + 2)^5$

Example 5: $y = \frac{2}{(9x + 3)^3}$

• Rewrite as $y = 2(9x + 3)^{-3}$

• Let $u = 9x + 3$

• $\frac{d}{dx}u^{-3} = -3u^{-4} \cdot u'$

• $y' = 2 \cdot (-3)(9x + 3)^{-4} \cdot 9 = -54(9x + 3)^{-4}$

Example 6: $y = \left(\frac{6x}{5x - 2}\right)^4$

• Let $u = \frac{6x}{5x - 2}$

• $\frac{d}{dx}u^4 = 4u^3 \cdot u'$

• Find $u'$ using the quotient rule:

• $u' = \frac{6(5x - 2) - 6x \cdot 5}{(5x - 2)^2} = \frac{30x - 12 - 30x}{(5x - 2)^2} = \frac{-12}{(5x - 2)^2}$

• $y' = 4\left(\frac{6x}{5x - 2}\right)^3 \cdot \frac{-12}{(5x - 2)^2}$

Example 7: $y = \sqrt{8x - 3} = (8x - 3)^{1/2}$

• Let $u = 8x - 3$

• $\frac{d}{dx}u^{1/2} = \frac{1}{2}u^{-1/2} \cdot u'$

• $y' = \frac{1}{2}(8x - 3)^{-1/2} \cdot 8 = 4(8x - 3)^{-1/2}$

Example 8: $y = \sec(x e^x)$

• Let $u = x e^x$

• $\frac{d}{dx}\sec(u) = \sec(u) \tan(u) \cdot u'$

• $u' = e^x + x e^x$

• $y' = \sec(x e^x) \tan(x e^x) (e^x + x e^x)$

Example 9: $y = \tan(\sin(8x + 1))$

• Let $u = \sin(8x + 1)$

• $\frac{d}{dx}\tan(u) = \sec^2(u) \cdot u'$

• $u' = \cos(8x + 1) \cdot 8$

• $y' = \sec^2(\sin(8x + 1)) \cdot \cos(8x + 1) \cdot 8$

Example 10: $y = \sin^3 x + \sin x^3$

• For $\sin^3 x$, use the chain rule: $3\sin^2 x \cos x$

• For $\sin x^3$, let $u = x^3$, so derivative is $\cos(x^3) \cdot 3x^2$

• $y' = 3\sin^2 x \cos x + \cos(x^3) \cdot 3x^2$

Example 11: $y = (5x - 1)^2 e^{4x}$

• This is a product of two functions; use the product rule:

• $(f \cdot g)' = f' g + f g'$

• $f = (5x - 1)^2$, $g = e^{4x}$

• $f' = 2(5x - 1) \cdot 5 = 10(5x - 1)$

• $g' = e^{4x} \cdot 4 = 4e^{4x}$

• $y' = 10(5x - 1) e^{4x} + 4(5x - 1)^2 e^{4x}$

Example 12: $y = \frac{\sin 3x}{2x - 1}$

• Use the quotient rule: $\left(\frac{f}{g}\right)' = \frac{f' g - f g'}{g^2}$

• $f = \sin 3x$, $g = 2x - 1$

• $f' = \cos 3x \cdot 3 = 3\cos 3x$

• $g' = 2$

• $y' = \frac{3\cos 3x (2x - 1) - 2\sin 3x}{(2x - 1)^2}$

Summary Table: Derivative Rules Used

Rule	Formula	When to Use
Power Rule	$\frac{d}{dx}x^n = n x^{n-1}$	Polynomials, powers of $x$
Chain Rule	$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$	Composite functions
Product Rule	$(fg)' = f'g + fg'$	Product of two functions
Quotient Rule	$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$	Quotient of two functions
Trigonometric Derivatives	See above	Trigonometric functions
Exponential Derivative	$\frac{d}{dx}e^x = e^x$	Exponential functions

Key Takeaways

• The chain rule is essential for differentiating composite functions.

• Always identify the inner and outer functions before applying the chain rule.

• Combine the chain rule with other rules (product, quotient) as needed for more complex expressions.

• Practice with a variety of examples to master the application of these rules.