Find `dy/dx if x^2-y^2=5`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x^2 - y^2 = 5\), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ x^2 - y^2 = 5 \]
Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(x^2) - \frac{d}{dx}(y^2) = \frac{d}{dx}(5) \]

### Step 2: Apply the differentiation rules
Using the power rule for \(x^2\) and the chain rule for \(y^2\):
\[ 2x - 2y \frac{dy}{dx} = 0 \]
Here, \(\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}\) because we treat \(y\) as a function of \(x\).

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

### Step 4: Solve for \(\frac{dy}{dx}\)
Dividing both sides by \(2y\):
\[ \frac{dy}{dx} = \frac{2x}{2y} = \frac{x}{y} \]

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

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