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 = √7x-1

Verified step by step guidance

1. Step 1: Identify the composite function structure. The given function is y = \(\sqrt{7x - 1}\). This can be seen as a composition of two functions: an inner function and an outer function.
2. Step 2: Define the inner function u = g(x). In this case, choose the expression inside the square root as the inner function: u = 7x - 1.
3. Step 3: Define the outer function y = f(u). The outer function is the square root function applied to u: y = \(\sqrt{u}\).
4. Step 4: Differentiate the outer function with respect to u. The derivative of y = \(\sqrt{u}\) with respect to u is \(\frac{dy}{du}\) = \(\frac{1}{2\sqrt{u}\)}.
5. Step 5: Differentiate the inner function with respect to x. The derivative of u = 7x - 1 with respect to x is \(\frac{du}{dx}\) = 7.

Key Concepts

Composite Functions

A composite function is formed when one function is applied to the result of another function. In the expression y = f(g(x)), g(x) is the inner function, and f(u) is the outer function. Understanding how to decompose a function into its inner and outer components is essential for differentiation and applying the chain rule.

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 found using the formula dy/dx = f'(g(x)) * g'(x). This rule allows us to compute the derivative of complex functions by breaking them down into simpler parts.

Inner and Outer Functions

Identifying the inner and outer functions is crucial for applying the chain rule effectively. In the given function y = √(7x - 1), the inner function can be defined as g(x) = 7x - 1, and the outer function as f(u) = √u. Recognizing these functions helps in calculating the derivatives accurately and understanding the structure of the composite function.