Find $\frac{d y}{d x}$
\[
e^{4 x}=\sin (x+2 y)
\]

Step 1 ：Differentiate both sides of the equation with respect to x. The derivative of \(e^{4x}\) with respect to x is \(4e^{4x}\).

Step 2 ：Use the chain rule to differentiate \(\sin(x+2y)\). The derivative of \(\sin(u)\) is \(\cos(u)\) times the derivative of u. Here, u = x + 2y, so we need to find the derivative of x + 2y with respect to x, which is 1 + 2\(\frac{dy}{dx}\). So, the derivative of the right side is \(\cos(x+2y)(1 + 2\frac{dy}{dx})\).

Step 3 ：Set these two derivatives equal to each other to get the equation we need to solve for \(\frac{dy}{dx}\).

Step 4 ：Solve the equation to find \(\frac{dy}{dx}\).

Step 5 ：The derivative \(\frac{d y}{d x}\) is given by \(\boxed{\frac{2e^{4x}}{\cos(x + 2y)} - \frac{1}{2}}\).