The Derivative of the Natural Logarithm

The derivative of the natural logarithm is defined the following way:

f(x) = ln (x)

f'(x)=\frac{1}{x}

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:

(1)  y_{1}=1x

(2)  y_{2}=ln(x)

The first derivative of equation (1) is 1, i.e. \frac{d y_{1}}{dx}=1. This means that for each possible x adding an additional x will increase the value of y_{1} by exactly one. You can see this in the graph below. The black line depicts y_{1}=1x. In contrary to y_{1}=1x is y_{2}=ln(x) not a constant function. y_{2}=ln(x) is depicted as the red line in the graph below. You can see that the slope of the logarithmic function decreases monotonically in x. The slope of y_{2}= ln(x) at a certain x is \frac{1}{x}. 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 x and remember that the derivative is simply \frac{1}{x}.

plot

 

 

Formal Derivation of the Derivative of the Natural Logarithm

We begin with

(1)  y=ln(x)

we rearrange to (by taking the inverse definition)

(2)  e^{y}=x

Take the derivative of both sides with respect to x.

(3)  e^{y} \frac{dy}{dx}=1

From (2) we know that e^{y}=x. We can substitute x in for e^{y} and get

(4)  x \frac{dy}{dx}=1

Rearranging gives us

(5)  \frac{dy}{dx}=\frac{1}{x}

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