What is the Quotient Rule and how is it used to find derivatives?

[Greg Bernhardt]

Definition/Summary

The quotient rule is a formula for the derivative of the quotient of two functions, for which derivatives exist.

Equations

$$f(x) = \frac{g(x)}{h(x)}$$

Then,

$$f'(x) = \frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}$$

here, $h(x) \: \neq \: 0$

Extended explanation

Even though a quotient can always be differentiated using the product and chain rules, it is easier and more efficient to remember and use the quotient rule.

Proof of the quotient rule:
$$f(x) = \frac{g(x)}{h(x)} = g(x)[h(x)]^{-1}$$
Using the product and chain rules:
$$f'(x) = g'(x)\:[h(x)]^{-1} - \: [h(x)]^{-2} \: h'(x)\:g(x)$$
and, putting this over a common denominator:
$$f'(x) = \frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}$$

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[fresh_42]

The quotient rule is just another version of the product or Leibniz rule:
$$(f\cdot g)'=f\,'\cdot g +f \cdot g'$$
The Leibniz rule appears in many variations:

• definition of a derivation: $D(ab)=D(a)b+aD(b)\text{ and }D([a,b)]=[D(a),b]+[a,D(b)]$
• integration by parts: $\int u'v = uv - \int uv'$
• quotient rule: $(f/g)' = (f\cdot g^{-1})'=f\,'g{-1}+(g^{-1})' f$
• Jacobi identity: $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$
• exterior derivatives: $d(f\alpha)=d(f\wedge \alpha)=df\wedge \alpha+f\wedge d\alpha$