Problem 5 - Calculus Early Transcendentals

Question

Let \(f(x)=\sin x .\) What is the value of \(f^{\prime}(\pi) ?\)

Step-by-Step Solution

Answer

Answer: The value of the derivative of the function f(x) = sin(x) at x = π is f'(π) = -1.

Step 1: Identify the function and the point where the derivative is to be evaluated

The function is given by \(f(x) = sin(x)\). We need to find the derivative \(f'(x)\) at \(x = \pi\).

Step 2: Differentiate the function with respect to x

Using the differentiation rules, we have: $$f'(x) = \frac{d}{dx}\sin(x) = \cos(x)$$

Step 3: Evaluate the derivative at the given point

We need to find the value of \(f'(\pi)\): $$f'(\pi) = \cos(\pi)$$

Step 4: Determine the value of the cosine function at π

Recall the value of cosine function at π: $$\cos(\pi) = -1$$

Step 5: Write the final answer

The value of the derivative \(f'(x)\) of the given function \(f(x)= \sin(x)\) at \(x = \pi\) is: $$f'(\pi) = -1$$