Product and Quotient Rules

Product Rule

If $f$ and $g$ are differentiable, then:

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

Memory Aid

"First times derivative of second, plus second times derivative of first"

Or using Leibniz notation:

$\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$

Quotient Rule

If $f$ and $g$ are differentiable and $g(x) \neq 0$:

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

Memory Aid

"Low d-high minus high d-low, over low squared"

Or using Leibniz notation:

$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

Extended Product Rule

For three functions:

$\frac{d}{dx}[f \cdot g \cdot h] = f' \cdot g \cdot h + f \cdot g' \cdot h + f \cdot g \cdot h'$

Examples

Example 1: Product Rule

Find the derivative of $f(x) = (x^2 + 1)(x^3 - 2x)$:

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

$= 3x^4 - 2x^2 + 3x^2 - 2 + 2x^4 - 4x^2$

$= 5x^4 - 3x^2 - 2$

Example 2: Quotient Rule

Find the derivative of $f(x) = \frac{x^2 + 1}{x - 3}$:

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

$= \frac{2x^2 - 6x - x^2 - 1}{(x - 3)^2}$

$= \frac{x^2 - 6x - 1}{(x - 3)^2}$

Example 3: Product with trig

Find $\frac{d}{dx}[x^2 \sin(x)]$:

$= x^2 \cdot \cos(x) + \sin(x) \cdot 2x = x^2 \cos(x) + 2x \sin(x)$

Example 4: Quotient with exponential

Find $\frac{d}{dx}\left[\frac{e^x}{x + 1}\right]$:

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

$= \frac{e^x(x + 1 - 1)}{(x + 1)^2}$

$= \frac{xe^x}{(x + 1)^2}$

Tips

1. Always simplify before differentiating if possible
2. For quotients like $\frac{1}{f(x)}$, you can use: $\frac{d}{dx}\left[\frac{1}{f}\right] = -\frac{f'}{f^2}$
3. Sometimes it's easier to rewrite as a product: $\frac{f}{g} = f \cdot g^{-1}$