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 = tan 5x²

Verified step by step guidance

1. Identify the composite function structure: The given function is y = \tan(5x^2). This can be expressed in the form y = f(g(x)), where g(x) is the inner function and f(u) is the outer function.
2. Define the inner function: Let u = g(x) = 5x^2. This represents the expression inside the tangent function.
3. Define the outer function: The outer function is y = f(u) = \tan(u). This represents the tangent of the inner function.
4. Differentiate the inner function: Find the derivative of u with respect to x, which is \frac{du}{dx} = \frac{d}{dx}(5x^2) = 10x.
5. Differentiate the outer function: Find the derivative of y with respect to u, which is \frac{dy}{du} = \sec^2(u).
6. Apply the chain rule: Use the chain rule to find \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}. Substitute the derivatives found in the previous steps: \frac{dy}{dx} = \sec^2(5x^2) \cdot 10x.

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 = tan(5x²) as y = f(g(x)), where g(x) is the inner function and f(u) is the outer function. Understanding how to identify 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, which is essential for solving the given problem.

Inner and Outer Functions

In a composite function, the inner function is the one that is applied first, while the outer function is applied to the result of the inner function. For the function y = tan(5x²), the inner function can be identified as g(x) = 5x² and the outer function as f(u) = tan(u). Recognizing these functions is key to correctly applying the chain rule and finding the derivative.