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

검증된 단계별 안내

Identify the inner function and the outer function: Let the inner function be \( u = g(x) = 5x^2 + 11x \) and the outer function be \( y = f(u) = u^{4/3} \).

Differentiate the inner function \( u = g(x) \) with respect to \( x \): \( \frac{du}{dx} = \frac{d}{dx}(5x^2 + 11x) \).

Differentiate the outer function \( y = f(u) \) with respect to \( u \): \( \frac{dy}{du} = \frac{d}{du}(u^{4/3}) \).

Apply the chain rule to find \( \frac{dy}{dx} \): \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

Substitute the expressions for \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the chain rule formula to express \( \frac{dy}{dx} \) in terms of \( x \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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 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.

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