3.4: The Product and Quotient Rules – Calculus Study Notes

3.4: The Product and Quotient Rules

Introduction

This section covers essential differentiation techniques in calculus: the Product Rule and Quotient Rule. These rules allow us to find derivatives of functions that are products or quotients of two differentiable functions. Mastery of these rules is fundamental for solving more complex calculus problems.

Sum and Difference Rule

Definition and Application

• Sum/Difference Rule: The derivative of a sum or difference of two functions is the sum or difference of their derivatives.

$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

• This rule is straightforward and forms the basis for more advanced differentiation techniques.

• Example: If $f(x) = x^2$ and $g(x) = \sin x$, then $\frac{d}{dx}[x^2 + \sin x] = 2x + \cos x$.

Product Rule

Definition

• The Product Rule is used to differentiate the product of two functions.

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

• Both functions must be differentiable.

• This rule is essential when neither function is constant.

Examples

• Example 1: $y = (8x^2 - 1)(2x - 5)$

  • Let $f(x) = 8x^2 - 1$, $g(x) = 2x - 5$

  • $f'(x) = 16x$, $g'(x) = 2$

  • Apply the product rule:

  $y' = (8x^2 - 1)'(2x - 5) + (8x^2 - 1)(2x - 5)'$

  $= 16x(2x - 5) + (8x^2 - 1) \cdot 2$

  $= 32x^2 - 80x + 16x^2 - 2$

  $= 48x^2 - 80x - 2$

• Example 2: $y = x^2 e^x - 17x$

  • Let $f(x) = x^2$, $g(x) = e^x$

  • $f'(x) = 2x$, $g'(x) = e^x$

  • Apply the product rule to $x^2 e^x$ and subtract the derivative of $17x$:

  $y' = (x^2)'e^x + x^2(e^x)' - (17x)'$

  $= 2x e^x + x^2 e^x - 17$

Quotient Rule

Definition

• The Quotient Rule is used to differentiate the quotient of two functions.

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Both $f(x)$ and $g(x)$ must be differentiable, and $g(x) \neq 0$.

Examples

• Example 3: $y = \frac{4x - 5}{x + 3}$

  • $f(x) = 4x - 5$, $g(x) = x + 3$

  • $f'(x) = 4$, $g'(x) = 1$

  • Apply the quotient rule:

  $y' = \frac{4(x + 3) - (4x - 5) \cdot 1}{(x + 3)^2}$

  $= \frac{4x + 12 - 4x + 5}{(x + 3)^2}$

  $= \frac{17}{(x + 3)^2}$

• Example 4: $y = \frac{7e^x}{3e^x + 8}$

  • $f(x) = 7e^x$, $g(x) = 3e^x + 8$

  • $f'(x) = 7e^x$, $g'(x) = 3e^x$

  • Apply the quotient rule:

  $y' = \frac{7e^x(3e^x + 8) - 7e^x \cdot 3e^x}{(3e^x + 8)^2}$

  $= \frac{21e^{2x} + 56e^x - 21e^{2x}}{(3e^x + 8)^2}$

  $= \frac{56e^x}{(3e^x + 8)^2}$

• Example 5: $y = (4t - 1)(7t + 7)^{-1}$

  • Rewrite as $y = \frac{4t - 1}{7t + 7}$

  • $f(x) = 4t - 1$, $g(x) = 7t + 7$

  • $f'(x) = 4$, $g'(x) = 7$

  • Apply the quotient rule:

  $y' = \frac{4(7t + 7) - (4t - 1)7}{(7t + 7)^2}$

  $= \frac{28t + 28 - 28t + 7}{(7t + 7)^2}$

  $= \frac{35}{(7t + 7)^2}$

• Example 6: $y = \frac{x - 6}{x e^x}$

  • $f(x) = x - 6$, $g(x) = x e^x$

  • $f'(x) = 1$, $g'(x) = e^x + x e^x$ (by product rule)

  • Apply the quotient rule:

  $y' = \frac{1 \cdot x e^x - (x - 6)(e^x + x e^x)}{(x e^x)^2}$

  $= \frac{x e^x - x e^x - x^2 e^x + 6 e^x + 6x e^x}{x^2 e^{2x}}$

  $= \frac{-x^2 e^x + 6 e^x + 6x e^x}{x^2 e^{2x}}$

Application: Tangent Line to a Curve

Finding the Equation of a Tangent Line

• To find the equation of the tangent line to $f(x)$ at $x = a$:

• 1. Compute $f'(a)$ (the slope at $x = a$).

• 2. Find $f(a)$ (the point of tangency).

• 3. Use the point-slope form: $y - f(a) = f'(a)(x - a)$.

Example: Find the equation of the tangent line to $f(x) = \frac{3x}{x + 1}$ at $x = 2$.

• Compute the derivative using the quotient rule:

$f'(x) = \frac{(3x)'(x + 1) - 3x \cdot (x + 1)'}{(x + 1)^2} = \frac{3(x + 1) - 3x \cdot 1}{(x + 1)^2} = \frac{3x + 3 - 3x}{(x + 1)^2} = \frac{3}{(x + 1)^2}$

• Evaluate at $x = 2$:

$f'(2) = \frac{3}{(2 + 1)^2} = \frac{3}{9} = \frac{1}{3}$

• Find $f(2)$:

$f(2) = \frac{3 \cdot 2}{2 + 1} = \frac{6}{3} = 2$

• Equation of the tangent line:

$y - 2 = \frac{1}{3}(x - 2)$ $y = \frac{1}{3}x - \frac{2}{3} + 2 = \frac{1}{3}x + \frac{4}{3}$

Summary Table: Differentiation Rules

Rule	Formula	When to Use
Sum/Difference Rule	$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$	Sum or difference of functions
Product Rule	$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$	Product of two functions
Quotient Rule	$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$	Quotient of two functions

Additional info: The examples provided illustrate step-by-step application of the product and quotient rules, including simplification and combining like terms. These rules are foundational for further topics in calculus, such as implicit differentiation and applications to physics and engineering problems.