Calculus How To

Derivative of ln (Natural Log), ln(kx), ln(x^2)

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What is a Natural Logarithm?

A natural logarithm (ln) is the inverse function of ex; It is a logarithm with base e (the base is always a positive number). In other words, y = ln x is the same thing as:

ey = x

This fact comes into play when we’re finding the derivative of the natural log.


It’s called the natural logarithm because of the “e” (Euler’s number). Mercator (1668) first used the term “natural” (in the Latin form log naturalis) for any logarithm to base e (as cited in O’Connore & Robertson, 2001).

What is the Derivative of ln?

The derivative of ln(x) or ln(kx) is 1/x. In notation, that’s:
derivative of lnx

The natural log function, and its derivative, is defined on the domain x > 0.

The derivative of ln(k), where k is any constant, is zero.

The second derivative of ln(x) is -1/x2. This can be derived with the power rule, because 1/x can be rewritten as x-1, allowing you to use the rule.


Derivative of ln: Steps

Watch this short (2 min) video to see how the derivative of ln is obtained using implicit differentiation. The video also shows how to calculate the derivative of ln(kx) and x2:


Can’t see the video? Click here.

To find the derivative of ln(x), use the fact that y = ln x can be rewritten as

ey = x
.
Step 1: Take the derivative of both sides of ey = x:
derivative of ln step 1

Step 2: Rewrite (using algebra) to get:
ln derivative step 2

Step 3: Substitute ln(x) for y:
derivative of natural log step

References

Adler, F. (2013). Modeling the Dynamics of Life: Calculus and Probability for Life Scientists. Cengage Learning.
Exponential Review.
Daugherty, Z. (2011). Derivatives of Exponential and Logarithm Functions.
O’Connor, J. & Robertson, E. (2001). The Number e. Retrieved August 20, 2020 from: https://mathshistory.st-andrews.ac.uk/HistTopics/e/#:~:text=In%201668%20Nicolaus%20Mercator%20published,for%20logarithms%20to%20base%20e.
Ping, X. (2016). Why natural constant “e” is called “natural”.

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Stephanie Glen. "Derivative of ln (Natural Log), ln(kx), ln(x^2)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/derivatives/derivative-of-ln-natural-log/
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