Derivative of the logarithm

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): $4.1.46$