AcademyDifferentiation
Academy
Quotient Rule
Level 1 - Math I (Physics) topic page in Differentiation.
Quotient Rule
Used to differentiate quotients of functions.
Quotient Rule
\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\]
Memory Aid
"Low d-high minus high d-low, over low squared"
Example
Find \(\frac{d}{dx}\left[\frac{x}{\sin(x)}\right]\)
Example
\[= \frac{1 \cdot \sin(x) - x \cdot \cos(x)}{[\sin(x)]^2}\]
Simplified
\[= \frac{\sin(x) - x\cos(x)}{\sin^2(x)}\]
Alternative: Product Rule
Sometimes it's easier to rewrite:
Alternative
\[\frac{d}{dx}\left[\frac{f}{g}\right] = f \cdot g^{-1}\]
Then use the chain rule with the product rule.