Second derivative of $$$\ln\left(x\right)$$$

The calculator will find the second derivative of $$$\ln\left(x\right)$$$, with steps shown.

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


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

The derivative of the natural logarithm is $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = {\color{red}\left(\frac{1}{x}\right)}$$

Thus, $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$.

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

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

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


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