Difference between revisions of "Derivative of the logarithm"
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Latest revision as of 04:59, 21 December 2017
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \log(z) = \dfrac{1}{z},$$ where $\log$ denotes the logarithm.
Proof
By the definition, $$\log(z) = \displaystyle\int_1^z \dfrac{1}{z} \mathrm{d}z.$$ Using the fundamental theorem of calculus, $$\dfrac{\mathrm{d}}{\mathrm{d}z} \log(z) = \dfrac{1}{z},$$ as was to be shown.
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.46$
