Find `dy/dx if y^2=sinx`

Answer

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

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ y^2 = \sin x \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Apply the chain rule on the left side
The left side is \(y^2\). When we differentiate \(y^2\) with respect to \(x\), we use the chain rule:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
This means we multiply by the derivative of \(y\) with respect to \(x\), which is \(\frac{dy}{dx}\).

### Step 3: Differentiate the right side
The right side is \(\sin x\). The derivative of \(\sin x\) with respect to \(x\) is:
\[ \frac{d}{dx}(\sin x) = \cos x \]

### Step 4: Set the derivatives equal to each other
Now we can set the derivatives from both sides equal to each other:
\[ 2y \frac{dy}{dx} = \cos x \]

### Step 5: Solve for \(\frac{dy}{dx}\)
To isolate \(\frac{dy}{dx}\), we divide both sides by \(2y\):
\[ \frac{dy}{dx} = \frac{\cos x}{2y} \]

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

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