Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = √u, u = sin x

검증된 단계별 안내

First, identify the functions involved: y = √u and u = sin(x). We need to find dy/dx using the chain rule.

The chain rule states that dy/dx = f'(u) * g'(x), where f(u) = √u and g(x) = sin(x).

Calculate f'(u): Since y = √u, f(u) = u^(1/2). The derivative f'(u) with respect to u is (1/2)u^(-1/2).

Calculate g'(x): Since u = sin(x), the derivative g'(x) with respect to x is cos(x).

Combine the derivatives using the chain rule: dy/dx = f'(u) * g'(x) = (1/2)u^(-1/2) * cos(x). Substitute u = sin(x) into the expression to get dy/dx = (1/2)(sin(x))^(-1/2) * cos(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.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if you have a function y = f(u) and u = g(x), then the derivative dy/dx is found by multiplying the derivative of f with respect to u, f'(u), by the derivative of u with respect to x, g'(x). This allows us to differentiate complex functions by breaking them down into simpler parts.

Derivative of Square Root Function

To differentiate a square root function, y = √u, we use the power rule. The square root can be expressed as u^(1/2), and its derivative is (1/2)u^(-1/2) or 1/(2√u). This derivative is crucial when applying the chain rule to composite functions involving square roots, as it provides the rate of change of the square root function with respect to its variable.

Derivative of Sine Function

The derivative of the sine function, u = sin x, is cos x. This is a basic derivative that is essential when applying the chain rule to functions involving trigonometric components. Understanding this derivative allows us to determine how the sine function changes with respect to x, which is necessary for calculating the overall derivative of composite functions like y = √u where u = sin x.

Derivative CalculationsIn Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).y = √u, u = sin x