In differential calculus, the "Chain Rule" says:

If $f(x)~=~g[h(x)]$ then

$f'(x)~=~g'[h(x)]h'(x)$

This is probably not a
good example for teaching,
as it might be difficult
for students to keep track
of which is which. It may
be better to mix functions
such as $\sin$ and powers to
make it easier to identify
$g$ versus $h.$

For example, $x^6$ can be thought of as $(x^2)^3,$ so:

$\frac{d}{dx}~x^6~=~\frac{d}{dx}~((x^2)^3)$

In this case:

$h(x)~=~x^2$ (the "inner" function) and,

$g(z)~=~z^3$ (the "outer" function, applied to $h(x)$ ).

Applying the product rule we get:

$\frac{d}{dx}~((x^2)^3)~=~[3(x^2)^2]\times(2x^1)$

The right hand side then simplifies to $3x^4\times2x$ which is $6x^5,$ as required.

The chain rule combined with the product rule allows us to derive the quotient rule.

Calculus DifferentiatingPolynomials DifferentiatingTrigFunctions IntegralCalculus IsaacNewton OrdinaryDifferentialEquation PartialDifferentialEquation ProductRule	(none)			(none)
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DifferentialCalculus QuotientRule			(none)
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		⇌	You are here ChainRule
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			Function ProductRule
↓	↓		↓	↓
Calculus CategoryMetaTopic DifferentiatingPolynomials DifferentiatingTrigFunctions IntegralCalculus	(none)			CoDomainOfAFunction DomainOfAFunction ImageOfAFunction Polynomial RangeOfAFunction RealNumber

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