Question

Let r(x) = f(g(h(x))), where h(1) = 2, g(2) = 4, h'(1) = 3, g'(2) = 3, and f '(4) = 6. Find r'(1). r'(1) =

Answer 1

To find r'(1) for the function r(x) = f(g(h(x))), we can use the chain rule by finding the derivatives h'(1), g'(2), and f'(4), and substituting them into the formula r'(1) = f'(4) * 3 * 3 = 54.

Step-by-step explanation:

We are given the function r(x)=f(g(h(x))), along with values for h(1), g(2), h'(1), g'(2), and f'(4). To find r'(1), we can use the chain rule. Let's start by finding the derivatives of h(1) and g(2): h'(1) = 3 and g'(2) = 3. Next, we'll find the derivative of f(g(h(x))):

f'(g(h(x))) = f'(g(h(1))) * g'(h(1)) * h'(1)

Substituting the given values, we have f'(4) * 3 * 3. Since h(x) is inside g(x), which is inside f(x), when we differentiate r(x), we multiply all these derivatives. Therefore, r'(x) = f'(4) * 3 * 3.

To find r'(1), we substitute x = 1 into the expression for r'(x):

r'(1) = f'(4) * 3 * 3 = 6 * 3 * 3 = 54.