Derivative of Natural Logarithmic Functions

A function defined by $y = {\log _a}x,\,\,\,x > 0$, where $x = {a^y},\,\,\,a > 0$, $a \ne 1$ is called the logarithm of $x$ to the base $a$. The natural logarithmic function is written as $y = {\log _e}x$ or $y = \ln x$.

We shall prove the formula for the derivative of the natural logarithm function using definition or the first principle method.

Let us suppose that the function is of the form

First we take the increment or small change in the function:

Putting the value of function $y = \ln x$ in the above equation, we get

Dividing both sides by $\Delta x$, we get

Multiplying and dividing the right hand side by $x$, we have

Taking the limit of both sides as $\Delta x \to 0$, we have

Consider $\frac{{\Delta x}}{x} = u \Rightarrow \frac{x}{{\Delta x}} = \frac{1}{u}$, as $\Delta x \to 0$ then $u \to 0$, we get

Using the relation from the limit $\mathop {\lim }\limits_{x \to 0} {\left( {1 + x} \right)^{\frac{1}{x}}} = e$, we have

Example: Find the derivative of

We have the given function as

Differentiating with respect to variable $x$, we get

Using the rule, $\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}$, we get