Editing Chain rule (section)

=== Quotient rule ===
{{See also|Quotient rule}}
The chain rule can be used to derive some well-known differentiation rules. For example, the quotient rule is a consequence of the chain rule and the [[product rule]]. To see this, write the function {{math|''f''(''x'')/''g''(''x'')}} as the product {{math|''f''(''x'') · 1/''g''(''x'')}}. First apply the product rule:
<math display="block">\begin{align}
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)
&= \frac{d}{dx}\left(f(x)\cdot\frac{1}{g(x)}\right) \\
&= f'(x)\cdot\frac{1}{g(x)} + f(x)\cdot\frac{d}{dx}\left(\frac{1}{g(x)}\right).
\end{align}</math>

To compute the derivative of {{math|1/''g''(''x'')}}, notice that it is the composite of {{mvar|g}} with the reciprocal function, that is, the function that sends {{mvar|x}} to {{math|1/''x''}}. The derivative of the reciprocal function is <math>-1/x^2\!</math>. By applying the chain rule, the last expression becomes:
<math display="block">f'(x)\cdot\frac{1}{g(x)} + f(x)\cdot\left(-\frac{1}{g(x)^2}\cdot g'(x)\right)
= \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2},</math>
which is the usual formula for the quotient rule.