What is `(dy)/(dx)` equal to ?

Options

A. `(y)/(x)`
B. `-(y)/(x)`
C. `(x)/(y)`
D. `-(x)/(y)`

Explanation

To find \(\frac{dy}{dx}\) for the equation of a circle given by \(x^2 + y^2 = a^2\), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the equation of the circle:**
\[ x^2 + y^2 = a^2 \]

2. **Differentiate both sides with respect to \(x\):**
- The derivative of \(x^2\) with respect to \(x\) is \(2x\).
- The derivative of \(y^2\) with respect to \(x\) requires the chain rule, which gives \(2y \frac{dy}{dx}\).
- The derivative of a constant \(a^2\) is \(0\).

Thus, we have:
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(a^2) \]
This simplifies to:
\[ 2x + 2y \frac{dy}{dx} = 0 \]

3. **Rearranging the equation:**
- Move \(2x\) to the right side:
\[ 2y \frac{dy}{dx} = -2x \]

4. **Solving for \(\frac{dy}{dx}\):**
- Divide both sides by \(2y\):
\[ \frac{dy}{dx} = \frac{-2x}{2y} \]
- Simplifying gives:
\[ \frac{dy}{dx} = \frac{-x}{y} \]

5. **Final result:**
\[ \frac{dy}{dx} = -\frac{x}{y} \]