Answer
$$\frac{dy}{dx}=\frac{2e^{2x}-\cos(x+3y)}{3\cos(x+3y)}$$
Work Step by Step
$$e^{2x}=\sin(x+3y)$$
1) Differentiate both sides of the equation with respect to $x$:
$$\frac{d}{dx}(e^{2x})=\frac{d}{dx}\sin(x+3y)$$
$$e^{2x}\frac{d}{dx}(2x)=\cos(x+3y)\frac{d}{dx}(x+3y)$$
$$2e^{2x}=\cos(x+3y)(1+3\frac{dy}{dx})$$
$$2e^{2x}=\cos(x+3y)+3\cos(x+3y)\frac{dy}{dx}$$
2) Collect all the terms with $dy/dx$ onto one side and solve for $dy/dx$:
$$3\cos(x+3y)\frac{dy}{dx}=2e^{2x}-\cos(x+3y)$$
$$\frac{dy}{dx}=\frac{2e^{2x}-\cos(x+3y)}{3\cos(x+3y)}$$
