Answer
\[\begin{align}
& \mathbf{a}\mathbf{.}\text{ }-3\pi \\
& \mathbf{b}\mathbf{.}\text{ }-5\pi \\
\end{align}\]
Work Step by Step
\[\begin{align}
& \text{Let }g\left( x \right)=\sin \left( \pi f\left( x \right) \right) \\
& \text{By the chain rule } \\
& g'\left( x \right)=\frac{d}{dx}\left[ \sin \left( \pi f\left( x \right) \right) \right] \\
& g'\left( x \right)=\cos \left( \pi f\left( x \right) \right)\frac{d}{dx}\left[ \pi f\left( x \right) \right] \\
& g'\left( x \right)=\pi \cos \left( \pi f\left( x \right) \right)\frac{d}{dx}\left[ f\left( x \right) \right] \\
& g'\left( x \right)=\pi \cos \left( \pi f\left( x \right) \right)f'\left( x \right) \\
& \\
& \mathbf{a}\mathbf{.}\text{ Calculate }g'\left( 0 \right) \\
& g'\left( 0 \right)=\pi \cos \left( \pi f\left( 0 \right) \right)f'\left( 0 \right) \\
& \text{Where }f\left( 0 \right)=-3\text{ and }f'\left( 0 \right)=3 \\
& g'\left( 0 \right)=\pi \cos \left( -3\pi \right)\left( 3 \right) \\
& g'\left( 0 \right)=\pi \left( -1 \right)\left( 3 \right) \\
& g'\left( 0 \right)=-3\pi \\
& \\
& \mathbf{b}\mathbf{.}\text{ Calculate }g'\left( 1 \right) \\
& g'\left( 1 \right)=\pi \cos \left( \pi f\left( 1 \right) \right)f'\left( 1 \right) \\
& \text{Where }f\left( 1 \right)=3\text{ and }f'\left( 1 \right)=5 \\
& g'\left( 1 \right)=\pi \cos \left( 3\pi \right)\left( 5 \right) \\
& g'\left( 1 \right)=\pi \left( -1 \right)\left( 5 \right) \\
& g'\left( 0 \right)=-5\pi \\
\end{align}\]
