Question

Let r(x) = f(g(h(x))), where h(1) = 4, g(4) = 5, h'(1) = 4, g'(4) = 4, and f '(5) = 8. Find r'(1).

Answer 1

To find r'(1), use the chain rule. First, find the derivatives of the individual functions. Then, apply the chain rule to calculate r'(1) by multiplying the derivatives together. The final answer is 32.

Step-by-step explanation:

To find r'(1), we need to use the chain rule. The chain rule states that if we have a function given as the composition of two or more functions, then the derivative of the composite function is the product of the derivatives of the individual functions.

First, we need to find the values of the derivatives of the functions involved:

• h'(1) = 4
• g'(4) = 4
• f '(5) = 8

Now, we can calculate r'(1) by using the chain rule:

1. Find the derivative of the outermost function: f'(g(h(x))) = f'(5) = 8
2. Find the derivative of the innermost function: g'(h(x)) = g'(4) = 4
3. Multiply the derivatives together: r'(x) = f'(g(h(x))) * g'(h(x)) = 8 * 4 = 32
4. Substitute x = 1 into the derivative: r'(1) = 32