\[\begin{align*} \boxed{\dfrac{d}{dx} \dfrac{d}{dx} \ln x = -\dfrac1{x^{-2}}} \end{align*}\]
Solving for the Derivative
To find the second derivative of $\ln x$, we must first find the first derivative of $\ln x$. The first derivative $\ln x$ is a common derivative, $\dfrac1x$. This is a derivative that you should memorize!
Now that we have the first derivative, we take the derivative of the first derivative, so we are finding:
\[\begin{align*} \dfrac{d}{dx} \left(\dfrac1x\right) \end{align*}\]
This is the same as $x^{-1}$, so we may apply the power rule:
\[\begin{align*} \dfrac{d}{dx} x^n = nx^{n-1} \end{align*}\]
Doing this with $x^{-1}$, we get our second derivative is equal to
\[\begin{align*} \dfrac{d}{dx} \dfrac{d}{dx} \ln x &=\dfrac{d}{dx} \left(\dfrac1x\right)\\ &=\dfrac{d}{dx} x^{-1}\\ &=-1 \cdot x^{-2}\\ &=\boxed{-\dfrac{1}{x^2}} \end{align*}\]
