5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = sin x⁵

Key Concepts

Composite Functions

A composite function is formed when one function is applied to the result of another function. In the context of the question, we express the function y = sin(x⁵) as a composition of two functions: an inner function g(x) = x⁵ and an outer function f(u) = sin(u). Understanding how to identify and separate these functions is crucial for applying the chain rule in differentiation.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if y = f(g(x)), then the derivative dy/dx can be calculated as dy/dx = f'(g(x)) * g'(x). This rule allows us to find the derivative of complex functions by breaking them down into simpler parts, making it essential for solving the given problem.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this problem, we need to differentiate the composite function y = sin(x⁵) using the chain rule. Understanding how to compute derivatives and apply differentiation techniques is vital for obtaining the final result of dy/dx.