Find the derivative of \(3x^{2} + \frac{1}{x^{2}}\)
Question 1 Report
Find the derivative of \(3x^{2} + \frac{1}{x^{2}}\)
Answer Details
To find the derivative of \(3x^{2} + \frac{1}{x^{2}}\), we will use the power rule and the chain rule of differentiation.
First, using the power rule, we can differentiate the term \(3x^{2}\) as follows:
\[\frac{d}{dx}(3x^{2}) = 6x.\]
Next, using the chain rule, we can differentiate the term \(\frac{1}{x^{2}}\) as follows:
\[\frac{d}{dx}\left(\frac{1}{x^{2}}\right) = -\frac{1}{x^{3}} \cdot \frac{d}{dx}(x^{-2}) = -\frac{1}{x^{3}} \cdot (-2x^{-3}) = \frac{2}{x^{5}}.\]
Putting these together, we get:
\[\frac{d}{dx}(3x^{2} + \frac{1}{x^{2}}) = 6x + \frac{2}{x^{5}}.\]
Therefore, the answer is option (C) \(6x - \frac{2}{x^{3}}\).
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