Question

If `f(x) =-e^(-x)` and `g(x) = tan^2x^2`, what is `f'(g(x))`?

Answer 1

`f' (g(x))=4x*e^(-tan^2 x^2)*(tan^3 x^2+tan x^2)`

Explanation:

given `f(x)=-e^(-x)` and `g(x)=tan^2 x^2`

`f(g(x))=-e^(-g(x))`

`f(g(x))=-e^(-tan^2 x^2)`

`f' (g(x))=(-1)*d/dx(e^(-tan^2 x^2))`

`f' (g(x))=(-1)*(e^(-tan^2 x^2))*d/dx(-tan^2 x^2)`

`f' (g(x))=(-1)*(e^(-tan^2 x^2))*-d/dx(tan^2 x^2)`

`f' (g(x))=(-1)*(e^(-tan^2 x^2))(-2(tan x^2)^(2-1))d/dx(tan x^2)`

`f' (g(x))=(-1)*(e^(-tan^2 x^2))(-2(tan x^2)^(2-1))(sec^2 x^2)d/dx(x^2)`

`f' (g(x))=(-1)*(e^(-tan^2 x^2))(-2(tan x^2)^(2-1))(sec^2 x^2)(2x)`

Simplify at this point

`f' (g(x))=(4x)*(e^(-tan^2 x^2))(tan x^2)(sec^2 x^2)`

`f' (g(x))=(4x)*(e^(-tan^2 x^2))(tan x^2)(tan^2 x^2+1)`

`f' (g(x))=(4x)*(e^(-tan^2 x^2))(tan^3 x^2+tan x^2)`

God bless....I hope the explanation is useful.