Question

find the derivative of (6x-x^3)^2

Answer 1

Chain rule states to take the derivative of the outer, multiply by the inner and then multiply by the derivative of the inner. With the equation simplified as such: (u)^n, chain rule states that the derivative will be (n)*((u)^(n-1))*(u')

Answer 2

If you have a composite function, that is: f(g(x)) then you need the chain rule or:
\[\frac{d}{dx}f(g(x))=f'(g(x))*g'(x)\]
So for this problem:
\[f(x)=x^2; g(x)=6x-x^3 \implies f(g(x))=(6x-x^3)^2\]
So we have:
\[f'(x)=2x \implies f'(g(x))=2(6x-x^3); g(x)=6x-x^3 \implies g'(x)=6-3x^2\]
\[\implies (f(g(x)))'=f'(g(x))g'(x)=2(6x-x^3)(6-3x^2)\]