Question

suppose f is differentiable for all real numbers with f(9) = 5 and f(9) = 7. let g(x) = sin(πf(x)). evaluate the following expression. g(9) g(9) = (type an exact answer, using π as needed)

Explanation:

Step1: Apply chain - rule

The chain - rule states that if \(y = \sin(u)\) and \(u=\pi f(x)\), then \(y^\prime=\cos(u)\cdot u^\prime\). So, \(g^\prime(x)=\cos(\pi f(x))\cdot\pi f^\prime(x)\).

Step2: Substitute \(x = 9\)

We know that \(f(9) = 5\) and \(f^\prime(9)=7\). Substitute \(x = 9\) into \(g^\prime(x)\):
\[

$$\begin{align*} g^\prime(9)&=\cos(\pi f(9))\cdot\pi f^\prime(9)\\ &=\cos(5\pi)\cdot\pi\cdot7 \end{align*}$$

\]
Since \(\cos(5\pi)=\cos(\pi + 4\pi)=- 1\), then \(g^\prime(9)=-7\pi\).

Answer:

\(-7\pi\)