Question

If `f(x)= 5x` and `g(x) = 3x^( 2/3 )`, what is `f'(g(x))`?

Answer 1

`f'(g(x))=color(green)(5)`
or
`f'(g(x))=color(green)(10/root(3)(x)`

perhaps depending upon the interpretation of `f'(g(x))`

Explanation:

Version 1
If `f(x)=5x`
then `f'(x)=5` (Exponent rule for derivatives)

That is `f'(x)` is a constant (`5`) for any value of `x`.

Specifically if we replace `x` with `g(x)`
`f'(g(x))` is still equal to the constant `5`

Version 2
If `f(x)=5x` and `g(x)=3x^(2/3)`
then `f(g(x)) =15x^2/3`
and
`f'(g(x)) = 2/3xx5x^(-1/3)=10/x^(1/3)=10/root(3)(x)`