Question

Find the value of (f ◦ g)' at the given value of x.

f(u) = sin^2 πu + u, u = g(x) = -x, x = 8

Answer 1

Chain rule:\[\frac{d}{dx}[(f\circ g)(x)]=\frac{d}{dx}[f(g(x))]=f'(g(x))\times g'(x)\]
In the form we're given, \(\dfrac{d}{dx}[(f\circ g)(x)]=f'(u)\times \dfrac{d}{dx}u=f'(u)\times g'(x)\).
In this case,
\[f(u)=\sin^2\pi u+u~~\Rightarrow~~f'(u)=\cdots\\
g(x)=-x~~~~~~~~~~~~~~\Rightarrow~~g'(x)=\cdots\]