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).
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
Step 1: Take the derivative of both sides of ey = x:
Step 2: Rewrite (using algebra) to get:
Step 3: Substitute ln(x) for y:
Adler, F. (2013). Modeling the Dynamics of Life: Calculus and Probability for Life Scientists. Cengage Learning.
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”.
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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