Study Notes: The Product and Quotient Rules in Calculus

Product and Quotient Rules

Introduction

The Product Rule and Quotient Rule are fundamental techniques in differential calculus used to compute derivatives of functions that are products or quotients of two differentiable functions. These rules extend the basic differentiation rules and are essential for handling more complex expressions.

Sum and Difference Rule

The Sum/Difference Rule states that the derivative of the sum or difference of two functions is the sum or difference of their derivatives.

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

• Example: If $f(x) = x^2$ and $g(x) = 3x$, then $(f(x) + g(x))' = 2x + 3$

Product Rule

The Product Rule is used when differentiating the product of two functions. If $f(x)$ and $g(x)$ are both differentiable, then:

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

• Example 1: $y = (8x^2 - 1)(2x - 5)$ $y' = (8x^2 - 1)' \cdot (2x - 5) + (8x^2 - 1) \cdot (2x - 5)'$ $= 16x \cdot (2x - 5) + (8x^2 - 1) \cdot 2$ $= 32x^2 - 80x + 16x^2 - 2$ $= 48x^2 - 80x - 2$

• Example 2: $y = x^2 e^x - 17x$ $y' = (x^2 e^x)' - (17x)'$ $= (x^2)' \cdot e^x + x^2 \cdot (e^x)' - 17$ $= 2x \cdot e^x + x^2 \cdot e^x - 17$

Quotient Rule

The Quotient Rule is used when differentiating the quotient of two functions. If $f(x)$ and $g(x)$ are both differentiable and $g(x) \neq 0$, then:

• Formula: $\left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}$

• Example 1: $y = \frac{4x - 5}{x + 3}$ $y' = \frac{(4x - 5)' \cdot (x + 3) - (4x - 5) \cdot (x + 3)'}{(x + 3)^2}$ $= \frac{4 \cdot (x + 3) - (4x - 5) \cdot 1}{(x + 3)^2}$ $= \frac{4x + 12 - 4x + 5}{(x + 3)^2}$ $= \frac{17}{(x + 3)^2}$

• Example 2: $y = \frac{7e^x}{3e^x + 8}$ $y' = \frac{(7e^x)' \cdot (3e^x + 8) - (7e^x) \cdot (3e^x + 8)'}{(3e^x + 8)^2}$ $= \frac{7e^x \cdot (3e^x + 8) - 7e^x \cdot 3e^x}{(3e^x + 8)^2}$ $= \frac{21e^{2x} + 56e^x - 21e^{2x}}{(3e^x + 8)^2}$ $= \frac{56e^x}{(3e^x + 8)^2}$

• Example 3: $y = \frac{4t - 1}{7t + 7}$ $y' = \frac{(4t - 1)' \cdot (7t + 7) - (4t - 1) \cdot (7t + 7)'}{(7t + 7)^2}$ $= \frac{4 \cdot (7t + 7) - (4t - 1) \cdot 7}{(7t + 7)^2}$ $= \frac{28t + 28 - 28t + 7}{(7t + 7)^2}$ $= \frac{35}{(7t + 7)^2}$

• Example 4: $y = \frac{x - 6}{x e^x}$ $y' = \frac{(x - 6)' \cdot (x e^x) - (x - 6) \cdot (x e^x)'}{(x e^x)^2}$ $= \frac{1 \cdot (x e^x) - (x - 6) \cdot (1 \cdot e^x + x \cdot e^x)}{x^2 e^{2x}}$ $= \frac{x e^x - x e^x - x^2 e^x + 6 e^x + 6x e^x}{x^2 e^{2x}}$ $= \frac{-x^2 e^x + 6 e^x + 6x e^x}{x^2 e^{2x}}$

Application: Tangent Line to a Curve

To find the equation of the tangent line to a curve at a specific point, first compute the derivative at that point to obtain the slope, then use the point-slope form of a line.

• Example: Find the tangent line to $f(x) = \frac{3x}{x + 1}$ at $x = 2$ $f'(x) = \frac{(3x)' \cdot (x + 1) - 3x \cdot (x + 1)'}{(x + 1)^2}$ $= \frac{3 \cdot (x + 1) - 3x \cdot 1}{(x + 1)^2}$ $= \frac{3x + 3 - 3x}{(x + 1)^2}$ $= \frac{3}{(x + 1)^2}$ At $ x = 2 $: $f'(2) = \frac{3}{9} = \frac{1}{3}$ $f(2) = \frac{6}{3} = 2$ Equation: $y - 2 = \frac{1}{3}(x - 2)$ $y = \frac{1}{3}x + \frac{4}{3}$

Summary Table: Product and Quotient Rules

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

*Additional info: Expanded explanations and step-by-step examples were added for clarity and completeness. The image provided is a university logo and is not directly relevant to the mathematical content, so it is not included.*