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 = (5x²+11x)^4/3

Verified step by step guidance

1. Identify the inner function and the outer function: Let the inner function be \( u = g(x) = 5x^2 + 11x \) and the outer function be \( y = f(u) = u^{4/3} \).
2. Differentiate the inner function \( u = g(x) \) with respect to \( x \): \( \frac{du}{dx} = \frac{d}{dx}(5x^2 + 11x) \).
3. Differentiate the outer function \( y = f(u) \) with respect to \( u \): \( \frac{dy}{du} = \frac{d}{du}(u^{4/3}) \).
4. Apply the chain rule to find \( \frac{dy}{dx} \): \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).
5. Substitute the expressions for \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the chain rule formula 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 given problem, we need to identify an inner function g(x) and an outer function f(u) such that the overall function can be expressed as y = f(g(x)). Understanding how to decompose a function into its components is essential for differentiation.

Chain Rule

The chain rule is a fundamental principle 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 crucial for solving the problem.

Power Rule

The power rule is a basic differentiation rule that states if y = x^n, then dy/dx = n*x^(n-1). In the context of the given function, which involves a power of a polynomial, applying the power rule to the outer function will be necessary to compute the derivative correctly. This rule simplifies the process of differentiation for polynomial expressions.