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 = e^4x²+1

Verified step by step guidance

1. Step 1: Identify the composite function structure. The given function is \( y = e^{4x^2 + 1} \). This can be seen as a composition of two functions.
2. Step 2: Define the inner function \( u = g(x) \). Here, choose \( u = 4x^2 + 1 \). This simplifies the expression inside the exponent.
3. Step 3: Define the outer function \( y = f(u) \). With \( u = 4x^2 + 1 \), the outer function becomes \( y = e^u \).
4. Step 4: Differentiate the outer function with respect to \( u \). The derivative of \( y = e^u \) with respect to \( u \) is \( \frac{dy}{du} = e^u \).
5. Step 5: Differentiate the inner function with respect to \( x \). The derivative of \( u = 4x^2 + 1 \) with respect to \( x \) is \( \frac{du}{dx} = 8x \). Now, use the chain rule to find \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^u \cdot 8x \). Substitute back \( u = 4x^2 + 1 \) to get \( \frac{dy}{dx} = e^{4x^2 + 1} \cdot 8x \).

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 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 inner and outer components is crucial 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 essential for solving the given problem.

Exponential Functions

Exponential functions are mathematical functions of the form y = a * e^(bx), where e is the base of natural logarithms. In the given function y = e^(4x² + 1), recognizing the structure of the exponential function is important for identifying the outer function. Understanding the properties of exponential functions, including their derivatives, is key to applying the chain rule effectively.