Quotient rule

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In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.^[1]^[2]^[3]

If the function one wishes to differentiate, $f(x)$ , can be written as

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

and $h(x)\not=0$ , then the rule states that the derivative of $g(x)/h(x)$ is

f'(x)={\frac {g'(x)h(x)-h'(x)g(x)}{[h(x)]^{2}}}.

Many people remember the Quotient Rule by the rhyme "Low D-high, High D-low, cross the line and square below." It is important to remember the 'D' describes the succeeding portion of the original fraction.

Proof using implicit differentiation

Let

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

Then

g(x)=f(x)h(x){\mbox{ }}\,

Using product rule,

g'(x)=f'(x)h(x)+f(x)h'(x){\mbox{ }}\,

f'(x)={\frac {g'(x)-f(x)h'(x)}{h(x)}}={\frac {g'(x)-{\frac {g(x)}{h(x)}}\cdot h'(x)}{h(x)}}

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

Proof using chain rule

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

We rewrite the fraction using a negative exponent.

f(x)=g(x)\cdot h(x)^{-1}

Take the derivative of both sides, and apply the product rule to the right side.

f'(x)=g'(x)\cdot h(x)^{-1}+g(x)\cdot (h(x)^{-1})'

To evaluate the derivative in the second term, apply the chain rule, where the outer function is $x^{-1}$ , and the inner function is $h(x)$ .

f'(x)=g'(x)\cdot h(x)^{-1}+g(x)\cdot (-1)h(x)^{-2}h'(x)

Rewrite things in fraction form.

f'(x)={\frac {g'(x)}{h(x)}}-{\frac {g(x)\cdot h'(x)}{[h(x)]^{2}}}

f'(x)={\frac {g'(x)\cdot h(x)-h'(x)\cdot g(x)}{[h(x)]^{2}}}

Higher order formulas

It is much easier to derive higher order quotient rules using implicit differentiation. For example, two implicit differentiations of $fh=g$ yields $f''h+2f'h'+fh''=g''$ and solving for $f''$ yields

f''={\frac {g''-2f'h'-fh''}{h}}.

References

↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
↑ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 0-547-16702-4.
↑ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 0-321-58876-2.

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