Derivative of Natural Logarithm 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 formula for derivative of natural logarithm function using by definition or first principle method.

Let us suppose that the function 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 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 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

Differentiation with respect to variable $x$, we get

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