对数函数导数公式推导过程

结论

$$
\frac{\rm d \ln x}{\rm d x}=\frac{1}{x}\rm d x
$$

$y=\ln x$的推导过程

$$
\begin{aligned}
\frac{d y}{d x} &=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} \
&=\lim _{\Delta x \rightarrow 0} \frac{\ln (x+\Delta x)-\ln (x)}{\Delta x} \
&=\lim _{\Delta x \rightarrow 0} \frac{\ln \left(1+\frac{\Delta x}{x}\right)}{\Delta x} \
&=\lim _{\Delta x \rightarrow 0} \frac{x}{x \Delta x} \ln \left(1+\frac{\Delta x}{x}\right) \
&=\lim {\Delta x \rightarrow 0} \frac{1}{x} \ln \left(1+\frac{\Delta x}{x}\right)^{\frac{x}{\Delta x}} \
&=\lim{\Delta x\rightarrow 0}\frac{1}{x} \ln e \
&=\frac{1}{x}
\end{aligned}
$$

$y=log_ax$的推导过程

$$
\begin{aligned}
&d \log _{a} x=d \frac{\ln x}{\ln a}=\frac{d \ln x}{\ln a}=\rm d\frac{1}{x \ln a}\
&d e^{x}=e^{x} d x\
&d a^{x}=d e^{\ln a^{x}}=d e^{x \ln a}=e^{x \ln a} d(x \ln a)=a^{x} \ln a d x
\end{aligned}
$$