$$\text { Find the value of } ( f \circ g ) ^ { \prime } \text { at the given value of } x \text {. }$$
-$$f ( u ) = \tan \frac { \pi u } { 2 } , u = g ( x ) = x ^ { 2 } , x = 4$$
A) 8
B) $- 4 \pi$
C) $4 \pi$
D) $- 16 \pi$

Question

Quizplus · Accepted Answer

To find the value of $(f \circ g)'$ at $x = 4$, we first need to find the derivative of the composite function $f(g(x))$, and then substitute $x = 4$. The composite function is $f(g(x)) = \tan\left(\frac{\pi x^2}{2}\right)$. The derivative of this function with respect to $x$ is found using the chain rule: $$\frac{d}{dx} \tan\left(\frac{\pi x^2}{2}\right) = \sec^2\left(\frac{\pi x^2}{2}\right) \cdot \frac{d}{dx}\left(\frac{\pi x^2}{2}\right).$$ The derivative of $\frac{\pi x^2}{2}$ with respect to $x$ is $\pi x$. Therefore, the derivative of the composite function is $$\sec^2\left(\frac{\pi x^2}{2}\right) \cdot \pi x.$$ Substituting $x = 4$ gives $$\sec^2\left(\frac{\pi \cdot 4^2}{2}\right) \cdot \pi \cdot 4 = \sec^2\left(8\pi\right) \cdot 4\pi.$$ Since $\sec^2(8\pi) = 1$ (because $\sec$ is the reciprocal of $\cos$, and $\cos(8\pi) = 1$), the final result is $4\pi$.

Find the value of (f∘g)′ at the given value of x. \text { Find the value of } ( f \circ g ) ^ { \prime } \text { at the given value of } x \text {. } Find the value of (f∘g)′ at the given value of x.

$\text { Find the value of } ( f \circ g ) ^ { \prime } \text { at the given value of } x \text {. }$