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:

$\frac{df(x)}{dx}=\frac{dg(x)}{dh(x)}*\frac{dh(x)}{dx}$

so that mean's we're analyzing:

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

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