Question

What is the instantaneous rate of change for f(x)=3^sin x at x=pi

Answer 1

-ln 3

Step-by-step explanation:

The instantaneous rate of change of f(x) at x = π is the derivative of this function at x = π.

We know that

`f(x)=3^(sinx)`

Then, the derivative is

`\begin{gathered} f^(\prime)(x)=3^(sinx)(\ln3)(\sin x)^(\prime) \\ f^(\prime)(x)=3^(sinx)(\ln3)\cos x \end{gathered}`

Now, we can replace x = π to get:

`\begin{gathered} f^(\prime)(\pi)=3^(\sin\pi)(\ln3)(\cos\pi) \\ f^(\prime)(\pi)=3^0(\ln3)(-1) \\ f^(\prime)(\pi)=(1)(\ln3)(-1) \\ f^(\prime)(\pi)=-\ln3 \end{gathered}`

Therefore, the instantaneous rate of change is -ln 3