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

Answer

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

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

### Step 2: Apply the chain rule
When differentiating \(y^2\), we use the chain rule. The derivative of \(y^2\) with respect to \(x\) is:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
The derivative of \(x\) with respect to \(x\) is simply:
\[ \frac{d}{dx}(x) = 1 \]

### Step 3: Set the derivatives equal
Now we set the derivatives equal to each other:
\[ 2y \frac{dy}{dx} = 1 \]

### Step 4: Solve for \(\frac{dy}{dx}\)
To isolate \(\frac{dy}{dx}\), we divide both sides by \(2y\):
\[ \frac{dy}{dx} = \frac{1}{2y} \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{1}{2y} \]

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