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.

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