The Quotient Rule


What is the quotient rule?

If f(x) and g(x) are differentiable and $g'(x)\neq0$ Then: $$\frac{d}{dx} \Big( \frac{f(x)}{g(x)} \Big)=\frac{g(x)f'(x)-f(x)g'(x)}{g^2(x)}$$
$$ \Big( \frac{f(x)}{g(x)} \Big)'=\frac{g(x)f'(x)-f(x)g'(x)}{g^2(x)}$$

Quotient Rule in Words

The denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all over the denominator squared.
Click to See

How can I remember this?

Here are two rhymes that might help you to memorize the quotient rule: $$(\frac{hi}{lo})'=\frac{(lo)(hi)'-(hi)(lo)' }{(lo)^2}$$ Think of "lo" as the denominator, "hi as the numerator" and "dee" as derivative:
" lo dee hi minus hi dee lo over the square of whats below "
"lo dee hi minus hi dee lo draw the line, and square below"
See Proof Of Quotient Rule

Example 2

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

Common Mistakes

  • Finding the derivatives in the wrong order in the numerator
  • Forgetting to divide by $(g(x))^2$
  • Adding in the numerator instead of subtracting