If $f (x) = tan 2 x$ $f(x)= tan2 x$ and $g (x) = \sqrt{- 4 x - 3}$ $g(x) = sqrt(-4x-3$ , how do you differentiate $f (g (x))$ $f(g(x))$ using the chain rule?

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Answer 1 · 2018-03-11T02:48:20

$- \frac{4 {sec}^{2} (2 \sqrt{- 4 x - 3})}{\sqrt{- 4 x - 3}}$ $-(4sec^2(2sqrt(-4x-3))) /sqrt(-4x-3)$

Starting with: $f (x) = tan 2 x$ $f(x) = tan2x$ and $g (x) = \sqrt{- 4 x - 3} = {(- 4 x - 3)}^{\frac{1}{2}}$ $g(x)=sqrt(-4x-3) = (-4x-3)^(1/2)$

Let $h (x) = f (g (x)) = tan (2 (\sqrt{- 4 x - 3}))$ $h(x) = f(g(x)) = tan(2(sqrt(-4x-3)))$

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

Using the Chain Rule:

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

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

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

If f(x)= tan2 x f(x)=tan2xf(x)= tan2 x and g(x) = sqrt(-4x-3 g(x)=√−4x−3g(x) = sqrt(-4x-3 , how do you differentiate f(g(x)) f(g(x))f(g(x)) using the chain rule?