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Second derivative of $$$\frac{1}{x}$$$
Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator
Your Input
Find $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x}\right)$$$.
Solution
Find the first derivative $$$\frac{d}{dx} \left(\frac{1}{x}\right)$$$
Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = -1$$$:
$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)} = {\color{red}\left(- \frac{1}{x^{2}}\right)}$$Thus, $$$\frac{d}{dx} \left(\frac{1}{x}\right) = - \frac{1}{x^{2}}$$$.
Next, $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x}\right) = \frac{d}{dx} \left(- \frac{1}{x^{2}}\right)$$$
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 $$$c = -1$$$ and $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \frac{1}{x^{2}}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}$$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, $$$\frac{d}{dx} \left(- \frac{1}{x^{2}}\right) = \frac{2}{x^{3}}$$$.
Therefore, $$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x}\right) = \frac{2}{x^{3}}$$$.
Answer
$$$\frac{d^{2}}{dx^{2}} \left(\frac{1}{x}\right) = \frac{2}{x^{3}}$$$A