Question

If `f(x)= - x^2 + x` and `g(x) = sqrtx + x`, how do you differentiate `f(g(x))` using the chain rule?

Answer 1

`-1-2sqrt(x)-x/sqrt(x)-2x`

Explanation:

Assuming you already know the chain rule. If not:
Given `h(x)` is a composite function consisting of 2 functions such that `h(x)=f(g(x))`, `h'(x)=f'(g(x))g'(x)`.

So, in this scenario, `f'(x)=-2x` and `g'(x)=1/(2(sqrt(x)))+1`
Substitute `g(x)` into `f'(x)`:
`-2(sqrt(x)+x)`
And multiply by `g'(x)`:
`-2(sqrt(x)+x)*(1/(2sqrt(x))+1)`
`-2(1/2+sqrt(x)+x/(2sqrt(x))+x)`
`-1-2sqrt(x)-x/sqrt(x)-2x`