Find the derivative of the following w.r.t. x :
`1/x-1/y-10=0`

Answer

To find the derivative of the equation \( \frac{1}{x} - \frac{1}{y} - 10 = 0 \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the equation
Start by rearranging the equation to isolate the terms involving \( y \):
\[ \frac{1}{x} - \frac{1}{y} = 10 \]

### Step 2: Differentiate both sides
Now, we will differentiate both sides of the equation with respect to \( x \). Remember that \( y \) is a function of \( x \), so we will use implicit differentiation:
\[ \frac{d}{dx}\left(\frac{1}{x}\right) - \frac{d}{dx}\left(\frac{1}{y}\right) = \frac{d}{dx}(10) \]

### Step 3: Apply the derivative
Using the derivative of \( \frac{1}{x} \) and applying the chain rule for \( \frac{1}{y} \):
\[ -\frac{1}{x^2} - \left(-\frac{1}{y^2} \cdot \frac{dy}{dx}\right) = 0 \]

### Step 4: Simplify the equation
This simplifies to:
\[ -\frac{1}{x^2} + \frac{1}{y^2} \cdot \frac{dy}{dx} = 0 \]

### Step 5: Solve for \(\frac{dy}{dx}\)
Rearranging the equation gives:
\[ \frac{1}{y^2} \cdot \frac{dy}{dx} = \frac{1}{x^2} \]
Now, multiply both sides by \( y^2 \):
\[ \frac{dy}{dx} = \frac{y^2}{x^2} \]

### Final Result
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{y^2}{x^2} \]

---