​Composition containing si​n​ x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.
c. g'(π)

Verified step by step guidance

1. Step 1: Recognize that g(x) is a composition of functions, where g(x) = f(sin x). To find g'(x), we need to use the chain rule for differentiation.
2. Step 2: The chain rule states that if you have a composition of functions h(x) = f(u(x)), then the derivative h'(x) = f'(u(x)) * u'(x).
3. Step 3: In this problem, u(x) = sin x, so u'(x) = cos x. Therefore, g'(x) = f'(sin x) * cos x.
4. Step 4: We need to evaluate g'(π). First, find sin(π), which is 0. So, g'(π) = f'(sin(π)) * cos(π).
5. Step 5: Substitute the known values: f'(0) = 3 and cos(π) = -1. Therefore, g'(π) = f'(0) * (-1).

Key Concepts

Chain Rule

The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. It states that if you have a function g(x) = f(h(x)), the derivative g'(x) can be found by multiplying the derivative of the outer function f with the derivative of the inner function h. This rule is essential for evaluating derivatives of functions that are composed of other functions.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In this context, sin(x) is used as the inner function in the composition g(x) = f(sin x). Understanding the behavior and derivatives of these functions is crucial for evaluating expressions involving them, especially at specific points like π.

Differentiability

A function is said to be differentiable at a point if it has a defined derivative there, which implies that the function is smooth and continuous at that point. In this problem, knowing that f is differentiable on the interval [−2,2] allows us to apply the Chain Rule confidently, as we can use the given derivatives f′(0) and f′(1) to find g'(π) through the composition of functions.