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

Explanation

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

### Step 1: Rewrite the function
We can rewrite the term \( \frac{1}{x^2} \) as \( x^{-2} \). Therefore, we can express the function as:
\[ y = x^2 + x^{-2} \]

### Step 2: Differentiate the function
Now, we will differentiate \( y \) with respect to \( x \). We will use the power rule for differentiation, 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 \( x^{-2} \):
\[ \frac{d}{dx}(x^{-2}) = -2 \cdot x^{-3} = -\frac{2}{x^3} \]

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

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

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