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

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

Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator

Leave empty for autodetection.
Leave empty, if you don't need the derivative at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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