Find `dy/dx if y^2=4ax`

Answer

To find \(\frac{dy}{dx}\) for the equation \(y^2 = 4ax\), we will use implicit differentiation. Here are the steps:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ y^2 = 4ax \]

2. **Differentiate both sides with respect to \(x\)**:
- The left side involves \(y\), which is a function of \(x\). We will use the chain rule to differentiate \(y^2\).
- The right side is a product of constants and \(x\).

Differentiating the left side:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]

Differentiating the right side:
\[ \frac{d}{dx}(4ax) = 4a \cdot \frac{d}{dx}(x) = 4a \]

3. **Set the derivatives equal to each other**:
\[ 2y \frac{dy}{dx} = 4a \]

4. **Solve for \(\frac{dy}{dx}\)**:
- To isolate \(\frac{dy}{dx}\), divide both sides by \(2y\):
\[ \frac{dy}{dx} = \frac{4a}{2y} \]

- Simplifying the right side gives:
\[ \frac{dy}{dx} = \frac{2a}{y} \]

### Final Result:
\[ \frac{dy}{dx} = \frac{2a}{y} \]