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 = sin⁵x

검증된 단계별 안내

Step 1: Identify the composite function structure. The given function is y = \(\sin\)^5(x), which can be rewritten as y = (\(\sin\)(x))^5.

Step 2: Determine the inner function u = g(x). In this case, the inner function is u = \(\sin\)(x).

Step 3: Determine the outer function y = f(u). The outer function is y = u^5.

Step 4: Differentiate the outer function with respect to u. The derivative of y = u^5 with respect to u is \(\frac{dy}{du}\) = 5u^4.

Step 5: Differentiate the inner function with respect to x. The derivative of u = \(\sin\)(x) with respect to x is \(\frac{du}{dx}\) = \(\cos\)(x).

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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 break down 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 differentiating the outer function and multiplying it by the derivative of the inner function.

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 given function y = sin⁵(x), recognizing that this is a power function of sin(x) is crucial. Applying the power rule in conjunction with the chain rule will help in finding the derivative of the composite function effectively.

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 = sin⁵x