Derivative Quotient Rule
Simple overview of the derivative quotient rule.
Derivative Quotient Rule
We use the derivative quotient rule when we can rewrite our function as the quotient of two functions.
\[f(x)=\frac{g(x)}{h(x)}\]Using this setup, the template for the quotient rule is
\[f'(x)=\frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}\]We can approach the quotient rule as a plug-in pattern:
- Identify the top and bottom functions.
- Find the derivatives of the top and bottom functions indvidually.
- Plug everything into the template to get the quotient rule.
Watch this section: general quotient rule template at 0:14.
Example 1: Polynomial Over Tangent
Consider
\[f(x)=\frac{x^5+3x-1}{\tan(x)}\]The top function ios
\[g(x)=x^5+3x-1\]and the bottom function is
\[h(x)=\tan(x)\]Now we find the top function’s derivative individually
\[g'(x)=5x^4+3\]and the bottom function’s derivative indidually
\[h'(x)=\sec^2(x)\]Substitute those four pieces into the quotient rule \(\left( f'(x)=\frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}\right)\)
\[f'(x)=\frac{\tan(x)(5x^4+3)-(x^5+3x-1)\sec^2(x)}{(\tan(x))^2}\]Watch this section: first quotient rule example at 0:32.
Example 2: Logarithm Over Sine
Consider
\[f(x)=\frac{\ln(x)}{\sin(x)}\]Here the top and bottom functions are
\[g(x)=\ln(x)\]and
\[h(x)=\sin(x)\]Their derivatives are
\[g'(x)=\frac{1}{x}\]and
\[h'(x)=\cos(x)\]Now plug those into the same quotient rule pattern \(\left( f'(x)=\frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}\right)\)
\[f'(x)=\frac{\sin(x)\left(\frac{1}{x}\right)-\ln(x)\cos(x)}{(\sin(x))^2}\]Watch this section: second quotient rule example at 2:10.
Timestamp Guide
Related Calculus Work
Use the quotient rule topic page for more quotient rule notes. The quotient rule example archive also collects timestamped quotient rule practice from JoeCMath videos.
For the surrounding derivative rules, browse the derivatives topic page or watch the JoeCMath derivative rules playlist: open the playlist on YouTube.