suppose f is differentiable for all real numbers with f(9) = 5 and f(9) = 7. let g(x) = sin(πf(x)). evaluate…

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)

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)

Answer

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)