If f(x)= tan2 x f(x)=tan2x and g(x) = sqrt(-4x-3 g(x)=4x3, how do you differentiate f(g(x)) f(g(x)) using the chain rule?

1 Answer
Mar 11, 2018

-(4sec^2(2sqrt(-4x-3))) /sqrt(-4x-3)4sec2(24x3)4x3

Explanation:

Starting with: f(x) = tan2xf(x)=tan2x and g(x)=sqrt(-4x-3) = (-4x-3)^(1/2)g(x)=4x3=(4x3)12

Let h(x) = f(g(x)) = tan(2(sqrt(-4x-3)))h(x)=f(g(x))=tan(2(4x3))

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)