Final answer:
To find rʹ(1), differentiate r(x) using the chain rule and substitute the given values. rʹ(1) = 120.
Step-by-step explanation:
Differentiation is a fundamental concept in calculus that deals with the rate at which a function changes. The derivative of a function measures how the output of the function changes with respect to changes in its input. If y=f(x) represents a function, then the derivative of f with respect to x is denoted by f′(x) or dx/dy. The process of finding derivatives is called differentiation.
To find rʹ(1), we need to differentiate r(x) with respect to x and then evaluate the derivative at x = 1. We'll start by finding the derivatives of each function.
Using the chain rule, we have:
rʹ(x) = fʹ(g(h(x))) × gʹ(h(x)) × hʹ(x)
Substituting the given values, we get:
rʹ(1) = fʹ(g(h(1))) × gʹ(h(1)) × hʹ(1)
= fʹ(6) × gʹ(5) × hʹ(1)
= 8 × 5 × 3 = 120