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Second derivative of $$$\frac{1}{x^{2}}$$$

The calculator will find the second derivative of $$$\frac{1}{x^{2}}$$$, with steps shown.

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Find $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x^{2}}\right)$$$.

Solution

Find the first derivative $$$\frac{d}{dx} \left(\frac{1}{x^{2}}\right)$$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = -2$$$:

$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)} = {\color{red}\left(- \frac{2}{x^{3}}\right)}$$

Thus, $$$\frac{d}{dx} \left(\frac{1}{x^{2}}\right) = - \frac{2}{x^{3}}$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x^{2}}\right) = \frac{d}{dx} \left(- \frac{2}{x^{3}}\right)$$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = -2$$$ and $$$f{\left(x \right)} = \frac{1}{x^{3}}$$$:

$${\color{red}\left(\frac{d}{dx} \left(- \frac{2}{x^{3}}\right)\right)} = {\color{red}\left(- 2 \frac{d}{dx} \left(\frac{1}{x^{3}}\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = -3$$$:

$$- 2 {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{3}}\right)\right)} = - 2 {\color{red}\left(- \frac{3}{x^{4}}\right)}$$

Thus, $$$\frac{d}{dx} \left(- \frac{2}{x^{3}}\right) = \frac{6}{x^{4}}$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x^{2}}\right) = \frac{6}{x^{4}}$$$.

Answer

$$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x^{2}}\right) = \frac{6}{x^{4}}$$$A