Calculus / Unit 4: Differentiation Rules
4.3 The Quotient Rule
Notes
For a quotient: $\displaystyle \frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$.
- The order matters: "low d-high minus high d-low, over low squared".
- Example: $\displaystyle \frac{d}{dx}\left[\frac{x}{x + 1}\right] = \frac{(1)(x + 1) - x(1)}{(x + 1)^2} = \frac{1}{(x + 1)^2}$.
- If the bottom is a single power of $x$, you can often rewrite and use the power rule instead.
Practice
Find $\displaystyle \frac{d}{dx}\left[\frac{x}{x + 1}\right]$.
$\displaystyle \frac{(x + 1) - x}{(x + 1)^2} = \frac{1}{(x + 1)^2}$.
Find $\displaystyle \frac{d}{dx}\left[\frac{x^2}{x - 2}\right]$.
$\displaystyle \frac{2x(x - 2) - x^2(1)}{(x - 2)^2} = \frac{x^2 - 4x}{(x - 2)^2}$.
Find $\displaystyle \frac{d}{dx}\left[\frac{3x + 2}{x - 1}\right]$.
$\displaystyle \frac{3(x - 1) - (3x + 2)(1)}{(x - 1)^2} = \frac{-5}{(x - 1)^2}$.
Ready for more? Take the Unit 4 test.