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The Quotient Rule

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What is the quotient rule?

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

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
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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 "
OR
"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