Composition containing sin x Suppose f is differentiable for all real numbers with f(0)=−3,f(1)=3,f′(0)=3, and f′(1)=5. Let g(x)=sin(πf(x)). Evaluate the following expressions.
b. g'(1)

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the derivative of the function g(x) = \(\sin\)(\(\pi\) f(x)) at x = 1. This involves using the chain rule for differentiation.
2. Step 2: Apply the chain rule. The chain rule states that if you have a composition of functions, such as g(x) = \(\sin\)(u) where u = \(\pi\) f(x), then the derivative g'(x) is given by g'(x) = \(\cos\)(u) \(\cdot\) u'(x).
3. Step 3: Differentiate the inner function u(x) = \(\pi\) f(x). The derivative u'(x) is \(\pi\) f'(x) because the derivative of a constant times a function is the constant times the derivative of the function.
4. Step 4: Substitute u = \(\pi\) f(x) and u'(x) = \(\pi\) f'(x) into the chain rule expression. This gives g'(x) = \(\cos\)(\(\pi\) f(x)) \(\cdot\) \(\pi\) f'(x).
5. Step 5: Evaluate g'(1) by substituting x = 1 into the expression. Use the given values f(1) = 3 and f'(1) = 5 to find g'(1) = \(\cos\)(\(\pi\) \(\cdot\) 3) \(\cdot\) \(\pi\) \(\cdot\) 5.

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) that is composed of another function f(x), the derivative g'(x) can be found by multiplying the derivative of the outer function g with respect to f by the derivative of the inner function f with respect to x. This is essential for evaluating g'(1) in the given problem.

Derivative of Trigonometric Functions

The derivative of trigonometric functions, such as sine, is crucial for solving problems involving these functions. Specifically, the derivative of sin(u) with respect to x is cos(u) multiplied by the derivative of u with respect to x. In this case, since g(x) involves sin(πf(x)), understanding how to differentiate sin with respect to its argument is necessary for finding g'(1).

Evaluating Derivatives at Specific Points

Evaluating derivatives at specific points involves substituting the given x-value into the derivative expression after it has been computed. In this problem, after applying the Chain Rule and finding g'(x), we will substitute x = 1 to find g'(1). This step is crucial for obtaining the final numerical result required by the question.