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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.