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

Answer

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

### Step 1: Differentiate both sides of the equation
Start with the given equation:
\[ x^2 + y^2 = r^2 \]
Differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(r^2) \]

### Step 2: Apply the differentiation rules
Using the power rule, we differentiate:
\[ 2x + 2y \frac{dy}{dx} = 0 \]
(Note that \(r^2\) is a constant, so its derivative is 0.)

### Step 3: Rearrange the equation
Now, isolate \(\frac{dy}{dx}\):
\[ 2y \frac{dy}{dx} = -2x \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Divide both sides by \(2y\):
\[ \frac{dy}{dx} = \frac{-2x}{2y} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{-x}{y} \]

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

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