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^√x

검증된 단계별 안내

Identify the composite function structure: The given function is y = e^{\(\sqrt{x}\)}. This can be seen as a composition of two functions.

Choose the inner function: Let u = g(x) = \(\sqrt{x}\). This represents the inner function, which is the square root of x.

Choose the outer function: Let y = f(u) = e^u. This represents the outer function, which is the exponential function with base e.

Differentiate the inner function: Find the derivative of u with respect to x, which is \(\frac{du}{dx}\) = \(\frac{1}{2\sqrt{x}\)}.

Differentiate the outer function: Find the derivative of y with respect to u, which is \(\frac{dy}{du}\) = e^u. Then, use the chain rule to find \(\frac{dy}{dx}\) = \(\frac{dy}{du}\) \(\cdot\) \(\frac{du}{dx}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 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 given problem.

Exponential Functions

Exponential functions are mathematical functions of the form y = a^x, where 'a' is a constant and 'x' is the variable. In the given question, the function involves the exponential function e raised to the power of another function, √x. Understanding the properties of exponential functions, including their derivatives, is vital for accurately calculating dy/dx in this context.

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^√x