Find `dy/dx if ax+by+c=0`

Answer

To find \(\frac{dy}{dx}\) from the equation \(ax + by + c = 0\), we will differentiate the equation with respect to \(x\). Here are the steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ ax + by + c = 0 \]

2. **Differentiate both sides with respect to \(x\):**
- The derivative of \(ax\) with respect to \(x\) is \(a\) (since \(a\) is a constant and the derivative of \(x\) is \(1\)).
- The derivative of \(by\) with respect to \(x\) is \(b \frac{dy}{dx}\) (using the chain rule, as \(y\) is a function of \(x\)).
- The derivative of \(c\) is \(0\) (since \(c\) is a constant).

Thus, differentiating gives:
\[ \frac{d}{dx}(ax) + \frac{d}{dx}(by) + \frac{d}{dx}(c) = 0 \]
This simplifies to:
\[ a + b \frac{dy}{dx} + 0 = 0 \]

3. **Rearranging the equation:**
\[ b \frac{dy}{dx} = -a \]

4. **Solve for \(\frac{dy}{dx}\):**
\[ \frac{dy}{dx} = -\frac{a}{b} \]

### Final Answer:
\[ \frac{dy}{dx} = -\frac{a}{b} \]

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