Find the derivative with respect to x of the following:
`(1+x^2)/( x^3)`

Answer

To find the derivative of the function \( y = \frac{1 + x^2}{x^3} \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function
First, we can separate the terms in the function:
\[ y = \frac{1}{x^3} + \frac{x^2}{x^3} \]
This simplifies to:
\[ y = \frac{1}{x^3} + \frac{1}{x} \]

### Step 2: Express in terms of negative exponents
Next, we can rewrite the fractions using negative exponents:
\[ y = x^{-3} + x^{-1} \]

### Step 3: Differentiate the function
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^{-3}) + \frac{d}{dx}(x^{-1}) \]
Using the power rule for differentiation, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \), we get:
\[ \frac{dy}{dx} = -3x^{-4} - 1x^{-2} \]

### Step 4: Simplify the derivative
We can rewrite the derivative in terms of positive exponents:
\[ \frac{dy}{dx} = -\frac{3}{x^4} - \frac{1}{x^2} \]

### Final Answer
Thus, the derivative of the function \( y = \frac{1 + x^2}{x^3} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{3}{x^4} - \frac{1}{x^2} \]

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