Question

Question #9eb92

Answer 1

Use the Power Rule and Chain Rule to write `h'(x)=4/5 (f(x))^{-1/5} * f'(x)=(4f'(x))/(5 root(5){f(x)})`.

Explanation:

The Power Rule says that `d/(dx) (x^{n})=nx^{n-1}` for any number `n`. The Chain Rule says that `d/(dx)(g(f(x)))=g'(f(x)) * f'(x)` when `f` is differentiable at the value `x` and `g` is differentiable at the value `f(x)`.

For the problem at hand, `g(x)=root(5){x^{4}}=x^{4/5}` and therefore `g'(x)=4/5 x^{-1/5}=4/(5 root(5){x})`.

Therefore `h'(x)=d/(dx)(g(f(x)))=4/5 (f(x))^{-1/5} * f'(x)=(4f'(x))/(5 root(5){f(x)})`