In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)

검증된 단계별 안내

Identify that the equation involves both \( x \) and \( y \) implicitly, so we will use implicit differentiation to find \( \frac{dy}{dx} \).

Differentiate both sides of the equation \( e^{2x} = \sin(x + 3y) \) with respect to \( x \). For the left side, use the chain rule: \( \frac{d}{dx} e^{2x} = 2e^{2x} \).

For the right side, apply the chain rule to \( \sin(x + 3y) \): \( \frac{d}{dx} \sin(x + 3y) = \cos(x + 3y) \cdot \frac{d}{dx}(x + 3y) \).

Differentiate \( x + 3y \) with respect to \( x \): \( \frac{d}{dx}(x + 3y) = 1 + 3 \frac{dy}{dx} \) because \( y \) is a function of \( x \).

Set up the equation: \( 2e^{2x} = \cos(x + 3y) (1 + 3 \frac{dy}{dx}) \). Then solve for \( \frac{dy}{dx} \) by isolating it on one side.

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.

Implicit Differentiation

Implicit differentiation is used when y is defined implicitly as a function of x, rather than explicitly. It involves differentiating both sides of an equation with respect to x, treating y as a function of x and applying the chain rule to terms involving y.

Chain Rule

The chain rule is a method for differentiating composite functions. When differentiating expressions like sin(x + 3y), you differentiate the outer function (sin) and multiply by the derivative of the inner function (x + 3y), applying the chain rule carefully to account for y as a function of x.

Solving for dy/dx

After differentiating implicitly, you collect all terms involving dy/dx on one side of the equation. Then, you solve algebraically for dy/dx to find the derivative of y with respect to x, which expresses the rate of change of y in terms of x and y.

In Exercises 27–32, find dy/dx.e^(2x)=sin(x+3y)