Find the derivative of `(1)/(x^(2))` from the first principle .

Options

A. `(-2)/(x)`
B. `2/(x^(3))`
C. `(-2)/(x^(2))`
D. `(-2)/(x^(3))`

Explanation

To find the derivative of \( y = \frac{1}{x^2} \) from the first principle, we will follow these steps:

### Step 1: Define the function
Let \( y = \frac{1}{x^2} \).

### Step 2: Increment the variable
Let \( \Delta x \) be a small increment in \( x \). Then the new value of \( x \) will be \( x + \Delta x \).

### Step 3: Find \( \Delta y \)
The corresponding increment in \( y \) is given by:
\[ \Delta y = f(x + \Delta x) - f(x) \]
Substituting \( f(x) = \frac{1}{x^2} \):
\[ \Delta y = \frac{1}{(x + \Delta x)^2} - \frac{1}{x^2} \]

### Step 4: Simplify \( \Delta y \)
To simplify \( \Delta y \), we can find a common denominator:
\[ \Delta y = \frac{x^2 - (x + \Delta x)^2}{x^2(x + \Delta x)^2} \]
Now, expand \( (x + \Delta x)^2 \):
\[ (x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2 \]
So,
\[ \Delta y = \frac{x^2 - (x^2 + 2x\Delta x + (\Delta x)^2)}{x^2(x + \Delta x)^2} \]
This simplifies to:
\[ \Delta y = \frac{-2x\Delta x - (\Delta x)^2}{x^2(x + \Delta x)^2} \]

### Step 5: Find \( \frac{\Delta y}{\Delta x} \)
Now, divide \( \Delta y \) by \( \Delta x \):
\[ \frac{\Delta y}{\Delta x} = \frac{-2x - \Delta x}{x^2(x + \Delta x)^2} \]

### Step 6: Take the limit as \( \Delta x \to 0 \)
Now, we will find the derivative \( \frac{dy}{dx} \) by taking the limit:
\[ \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{-2x - \Delta x}{x^2(x + \Delta x)^2} \]
As \( \Delta x \to 0 \), this becomes:
\[ \frac{dy}{dx} = \frac{-2x}{x^2(x)^2} = \frac{-2x}{x^4} = -\frac{2}{x^3} \]

### Final Answer
Thus, the derivative of \( \frac{1}{x^2} \) is:
\[ \frac{dy}{dx} = -\frac{2}{x^3} \]