Find `dy/dx if tan2x=siny`

Answer

To find \(\frac{dy}{dx}\) given the equation \(\tan(2x) = \sin(y)\), we will use implicit differentiation. Here are the steps to solve the problem:

### Step 1: Differentiate both sides with respect to \(x\)

We start with the equation:
\[ \tan(2x) = \sin(y) \]

Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}[\tan(2x)] = \frac{d}{dx}[\sin(y)] \]

### Step 2: Apply the chain rule

Using the chain rule, we differentiate the left side:
\[ \frac{d}{dx}[\tan(2x)] = \sec^2(2x) \cdot \frac{d}{dx}[2x] = \sec^2(2x) \cdot 2 \]

For the right side, we also apply the chain rule:
\[ \frac{d}{dx}[\sin(y)] = \cos(y) \cdot \frac{dy}{dx} \]

### Step 3: Set the derivatives equal

Now we can set the derivatives equal to each other:
\[ 2 \sec^2(2x) = \cos(y) \cdot \frac{dy}{dx} \]

### Step 4: Solve for \(\frac{dy}{dx}\)

To isolate \(\frac{dy}{dx}\), we rearrange the equation:
\[ \frac{dy}{dx} = \frac{2 \sec^2(2x)}{\cos(y)} \]

### Final Result

Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{2 \sec^2(2x)}{\cos(y)} \]