Question

How do you differentiate `f(x)=e^(x-(x-2)^2` using the chain rule.?

Answer 1

`f'(x) = (5 - 2x)e^(-x^2 + 5x - 4)`

Explanation:

Expand:

`f(x) = e^(x - (x^2 - 4x + 4))`

`f(x) = e^(x - x^2 + 4x - 4)`

`f(x) = e^(-x^2 + 5x - 4)`

We now use the chain rule to differentiate. Let `y = e^u` and `u = -x^2 + 5x - 4`. Then `y' = e^u` and `u' = -2x + 5`.

`f'(x) = u' * y'`

`f'(x) = e^u * 2x + 5`

`f'(x) = (5 - 2x)e^(-x^2 + 5x - 4)`

Hopefully this helps!