对数求导

\[{\displaystyle {\begin{aligned}{\frac \ln x}x}}&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}({\frac {1}{h}}\ln({\frac {x+h}{x}}))\\&=\lim _{h\to 0}({\frac {x}{xh}}\ln(1+{\frac {h}{x}}))\\&={\frac {1}{x}}\ln(\lim _{h\to 0}(1+{\frac {h}{x}})^{\frac {x}{h}})\\&={\frac {1}{x}}\ln e\\&={\frac {1}{x}}\end{aligned}}}\] \[{\frac \log _{\alpha }|x|}x}}={1 \over \ln \alpha }{\frac \ln |x|}x}}={1 \over x\ln \alpha }\]

References

  1. Differentiation rules