`y=x^(2)+(1)/(x^(2))`. Find `(dy)/(dx)`

Explanation

To find the derivative of the function \( y = x^2 + \frac{1}{x^2} \) with respect to \( x \), we will apply the rules of differentiation step by step.

### Step 1: Identify the function
We have:
\[ y = x^2 + \frac{1}{x^2} \]

### Step 2: Rewrite the function
The term \( \frac{1}{x^2} \) can be rewritten using negative exponents:
\[ y = x^2 + x^{-2} \]

### Step 3: Differentiate each term
Now we will differentiate each term separately using the power rule, which states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \).

1. Differentiate \( x^2 \):
\[ \frac{d}{dx}(x^2) = 2x \]

2. Differentiate \( x^{-2} \):
\[ \frac{d}{dx}(x^{-2}) = -2x^{-3} \]

### Step 4: Combine the derivatives
Now we can combine the results from the differentiation:
\[ \frac{dy}{dx} = 2x - 2x^{-3} \]

### Step 5: Rewrite the result
For clarity, we can rewrite \( -2x^{-3} \) back into fraction form:
\[ \frac{dy}{dx} = 2x - \frac{2}{x^3} \]

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

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