Question

How do you differentiate `f(x)=(4x-x^2)^100`?

Answer 1

`f'(x) = 100(4x-x^2)^99 (4-2x)`

Explanation:

Let `u=4x-x^2` and then `f(x)=u^100`

`(du)/dx = 4-2x`

Then, `f'(x)=d/dx(f(x))`
`:. f'(x) = d/dx u^100`
`:. f'(x) = d/(du) u^100 (du)/dx` (the chain rule)
`:. f'(x) = 100u^99 (4-2x)`
`:. f'(x) = 100(4x-x^2)^99 (4-2x)`