Question

If `f(x)= tan2 x` and `g(x) = sqrt(-4x-3`, how do you differentiate `f(g(x))` using the chain rule?

Answer 1

#-(4sec^2(2sqrt(-4x-3))) /sqrt(-4x-3)#

Explanation:

Starting with: `f(x) = tan2x` and `g(x)=sqrt(-4x-3) = (-4x-3)^(1/2)`

Let `h(x) = f(g(x)) = tan(2(sqrt(-4x-3)))`

`f'(x) = 2sec^2(2x)`
`g'(x) = (1/2)(-4)(-4x-3)^(-1/2) = -2/sqrt(-4x-3)`

Using the Chain Rule:

`h'(x) = f'(g(x))*g'(x)`

`h'(x) = 2sec^2(2sqrt(-4x-3))*-2/sqrt(-4x-3)`

`h'(x) = -(4sec^2(2sqrt(-4x-3)))/sqrt(-4x-3)`