Answer
\[\begin{align*} \boxed{\frac{d}{dx}(\ln x) = \frac{1}{x}} \end{align*}\]
The Derivative of \(\ln x\)
The function $\ln x$ is called the natural logarithm function. More specifically, $\ln x = \ln_e x$, where $e \approx 2.718$.
The derivative of $\ln x$ is:
\[\begin{align*} \frac{d}{dx}(\ln x) = \frac{1}{x} \end{align*}\]
However, some sources also use $\log x$ to describe $\ln x$. The reader may be more used to $\log x$ being the common logarithm $\log_{10} x$. In this case, the derivative of $\log x$ is
\[\begin{align*} \frac{d}{dx}(\log_{10} x) = \dfrac{1}{x \ln (10)} \end{align*}\]
It is important not to get confused when a textbook uses $\log x$ to designate the natrual logarithm. In order to stay consistent, we will define $\ln x$ as the natural logarithm in this article.
Example
Let’s look at a simple example. Suppose we want to find the rate of change of $\ln x$ at $x = 5$. Since we are finding rate of change, we take the derivative of $\ln x$:
\[\begin{align*} \dfrac{d}{dx}(\ln x) = \dfrac{1}{x}. \end{align*}\]
Since $x = 5$, we substitute this value of $x$ into the derivative, yielding $\dfrac{1}{5}$.
If we wanted to find the rate of change at any other value of $x$, we would have only needed to substitute into $\dfrac{1}{x}$. Notice that we cannot plug in $x = 0$, as this would result in division by zero. This means that the rate of change of $\ln x$ at $x = 0$ is undefined. For more information on derivatives and logarithms, check out other blogs on iacedcalculus.com.
