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

Answer

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Answer

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

Steps

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

Step 2 ：Use the chain rule to differentiate $\sin (x + 2 y)$ . 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 $\frac{d y}{d x}$ . So, the derivative of the right side is $\cos (x + 2 y) (1 + 2 \frac{d y}{d x})$ .

Step 3 ：Set these two derivatives equal to each other to get the equation we need to solve for $\frac{d y}{d x}$ .

Step 4 ：Solve the equation to find $\frac{d y}{d x}$ .

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

Find dydxe4x=sin⁡(x+2y)

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