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

Answer

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

### Step 1: Rewrite the function
We start with the function:
\[ y = 3x^2 + \frac{3}{x^2} \]
We can rewrite \( \frac{3}{x^2} \) as \( 3x^{-2} \) for easier differentiation:
\[ y = 3x^2 + 3x^{-2} \]

### Step 2: Differentiate each term
Now, we differentiate each term with respect to \( x \):
1. The derivative of \( 3x^2 \):
\[ \frac{d}{dx}(3x^2) = 6x \]
2. The derivative of \( 3x^{-2} \):
\[ \frac{d}{dx}(3x^{-2}) = 3 \cdot (-2)x^{-3} = -\frac{6}{x^3} \]

### Step 3: Combine the derivatives
Now, we combine the derivatives from Step 2:
\[ \frac{dy}{dx} = 6x - \frac{6}{x^3} \]

### Final Answer
Thus, the derivative of the function \( y = 3x^2 + \frac{3}{x^2} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 6x - \frac{6}{x^3} \]

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