Derivative Quotient Rule
This page follows the JoeCMath video Derivative Quotient Rule: 2 Examples, One EASY Pattern. The video uses two worked derivatives to show how the quotient rule works when one function is divided by another.
Quotient Rule Template
The video starts by writing a quotient as
\[f(x)=\frac{g(x)}{h(x)}.\]For that setup, the quotient rule is
\[f'(x)=\frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}.\]The video keeps the quotient rule as a plug-in pattern: identify the top, identify the bottom, then use the derivatives in the rule. The denominator function $h(x)$ appears twice: once multiplying $g’(x)$ in the numerator, and again squared in the denominator.
Watch this section: general quotient rule template at 0:14.
Example 1: Polynomial Over Tangent
The first example is
\[f(x)=\frac{x^5+3x-1}{\tan(x)}.\]In the video’s notation, the top function is
\[g(x)=x^5+3x-1,\]and the bottom function is
\[h(x)=\tan(x).\]Now take each derivative:
\[g'(x)=5x^4+3\]and
\[h'(x)=\sec^2(x).\]Substitute those four pieces into the quotient rule:
\[f'(x)=\frac{\tan(x)(5x^4+3)-(x^5+3x-1)\sec^2(x)}{(\tan(x))^2}.\]In the first example, the denominator is not just copied into the final denominator; it is squared. The numerator also keeps the order from the rule: bottom times derivative of the top, minus top times derivative of the bottom.
Watch this section: first quotient rule example at 0:32.
Example 2: Logarithm Over Sine
The second example is
\[f(x)=\frac{\ln(x)}{\sin(x)}.\]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:
\[f'(x)=\frac{\sin(x)\left(\frac{1}{x}\right)-\ln(x)\cos(x)}{(\sin(x))^2}.\]The second example follows the same pattern, but the derivatives are the familiar pieces from logarithms and trigonometry. Once $g(x)$, $h(x)$, $g’(x)$, and $h’(x)$ are identified, the rest of the work is substitution into the rule.
Watch this section: second quotient rule example at 2:10.
Closing Pattern
The closing idea of the video is that the quotient rule becomes much easier when the setup is organized before any substituting happens.
For a quotient
\[f(x)=\frac{g(x)}{h(x)},\]write down
\[g(x),\qquad h(x),\qquad g'(x),\qquad h'(x)\]first. Then place those pieces into
\[f'(x)=\frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}.\]That is the pattern both examples use.
Watch this section: closing reminder at 3:14.
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.