Question

How do I differentiate the function f(x)=(-2x^3+1)^5 with respect to x?

Answer 1

To solve this problem, we're going to have to use the chain rule. Let
`g(x)=x^5 \text{ and } h(x)=-2x^3+1`

f(x)=g(h(x)) which means that g(x)'s input changes at a different speed than x and that speed is the derivative of h(x). Written down, since it seems kinda weird to explain, we have:

`(df(x))/(dx)=(dg(x))/(dh(x))*(dh(x))/(dx)`

so that mean's we're analyzing:

`f'(x)=g'(h(x)))h'(x)`

So:
`5(-2x^3+1)^4*(-6x^2)=-480 x^(14) + 960 x^(11) - 720 x^(8) + 240 x^(5) - 30 x^2`