The derivative of the natural logarithm is defined the following way:
The formal proof of the derivative is provided at the bottom of this post.
The following example further explains the derivative of the natural logarithm. Remember that the derivative of a function gives you the slope of that very function, i.e. the first derivative gives you the slope of the function on a given point of evaluation. For instance, assume you have the following two functions:
The first derivative of equation (1) is 1, i.e. . This means that for each possible adding an additional will increase the value of by exactly one. You can see this in the graph below. The black line depicts . In contrary to is not a constant function. is depicted as the red line in the graph below. You can see that the slope of the logarithmic function decreases monotonically in . The slope of at a certain is . Next time you do not remember the derivative of the natural logarithm just remember its functional form. You will remember that the slop decreases with an increasing and remember that the derivative is simply .
Formal Derivation of the Derivative of the Natural Logarithm
We begin with
we rearrange to (by taking the inverse definition)
Take the derivative of both sides with respect to x.
From (2) we know that . We can substitute in for and get
Rearranging gives us