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 = (3x+7)¹⁰

Verified step by step guidance

1. Step 1: Identify the composite function structure. The given function is \( y = (3x+7)^{10} \). This is a composite function where an inner function is raised to a power.
2. Step 2: Define the inner function \( u = g(x) \). Here, choose \( u = 3x + 7 \) as the inner function.
3. Step 3: Define the outer function \( y = f(u) \). With \( u = 3x + 7 \), the outer function becomes \( y = u^{10} \).
4. Step 4: Differentiate the outer function with respect to \( u \). The derivative \( \frac{dy}{du} \) of \( y = u^{10} \) is \( 10u^9 \).
5. Step 5: Differentiate the inner function with respect to \( x \). The derivative \( \frac{du}{dx} \) of \( u = 3x + 7 \) is \( 3 \). Now, use the chain rule to find \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 10u^9 \cdot 3 \). Substitute back \( u = 3x + 7 \) to express \( \frac{dy}{dx} \) in terms of \( x \).

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 = (3x + 7)¹⁰ as a composition of two functions: an inner function g(x) = 3x + 7 and an outer function f(u) = u¹⁰. Understanding how to identify and separate these functions is crucial for 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 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, which is 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 = (3x + 7)¹⁰ using the Chain Rule. Understanding how to apply differentiation techniques is vital for calculating dy/dx accurately.