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

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

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