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

Verified step by step guidance

1. Identify that the equation involves both \( x \) and \( y \) implicitly, so we will use implicit differentiation to find \( \frac{dy}{dx} \).
2. Differentiate both sides of the equation \( e^{2x} = \sin(x + 3y) \) with respect to \( x \). For the left side, use the chain rule: \( \frac{d}{dx} e^{2x} = 2e^{2x} \).
3. For the right side, apply the chain rule to \( \sin(x + 3y) \): \( \frac{d}{dx} \sin(x + 3y) = \cos(x + 3y) \cdot \frac{d}{dx}(x + 3y) \).
4. Differentiate \( x + 3y \) with respect to \( x \): \( \frac{d}{dx}(x + 3y) = 1 + 3 \frac{dy}{dx} \) because \( y \) is a function of \( x \).
5. 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.

Key Concepts

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.