七则运算微分公式的推导

以下 \(c\) 为常数，\(x\) 为参数，\(e\) 为自然常数

部分表述不一定严谨

加法法则

\[(f(x)+g(x))^{'}=f^{'}(x)+g^{'}(x) \]

乘法法则

\[\begin{aligned} (f(x)g(x))^{'}=&\frac{f(x+\mathrm dx)g(x+\mathrm dx)-f(x)g(x)}{\mathrm dx}\\ =&\frac{(f(x)+f^{'}(x)\mathrm dx)(g(x)+g^{'}(x)\mathrm dx)-f(x)g(x)}{\mathrm dx}\\ =&\frac{f(x)g(x)+g(x)f^{'}(x)\mathrm dx+f(x)g^{'}(x)\mathrm dx+f^{'}(x)g^{'}(x)(\mathrm dx)^2-f(x)g(x)}{\mathrm dx}\\ =&\frac{g(x)f^{'}(x)\mathrm dx+f(x)g^{'}(x)\mathrm dx}{\mathrm dx}\\ =&g(x)f^{'}(x)+f(x)g^{'}(x)\\ \end{aligned}\]

链式法则

\[\begin{aligned} (f(g(x)))^{'}=&\frac{f(g(x+\mathrm dx))-f(g(x))}{\mathrm dx}\\ =&\frac{f(g(x)+g^{'}(x)\mathrm dx)-f(g(x))}{\mathrm dx}\\ =&\frac{f(g(x))+f^{'}(g(x))g^{'}(x)\mathrm dx-f(g(x))}{\mathrm dx}\\ =&\frac{f^{'}(g(x))g^{'}(x)\mathrm dx}{\mathrm dx}\\ =&f^{'}(g(x))g^{'}(x)\\ \end{aligned}\]

指数函数求导

\(e\) 的一种定义：令 \(f(x)=a^x\)，\(e\) 为使 \(f'(0)=1\) 的常数 \(a\)

\[\begin{aligned} \exp^{'}(x)=&\frac{\exp(x+\mathrm dx)-\exp(x)}{\mathrm dx}\\ =&\exp(x)\cdot\frac{\exp(\mathrm dx)-1}{\mathrm dx}\\ =&\exp(x)\cdot(\exp^{'}(x)[0])\\ =&\exp(x)\\ \\ (\exp(f(x)))^{'}=&\exp^{'}(f(x))f^{'}(x)\\ =&\exp(f(x))f^{'}(x)\\ \\ \left(c^{x}\right)^{'}=&\exp^{'}(x\ln c)\\ =&\exp(x\ln c)(x\ln c)^{'}\\ =&\exp(x\ln c)\ln c\\ =&c^x\ln c\\ \end{aligned}\]

自然对数求导

\[\begin{aligned} e^{\ln x}&=x\\ (e^{\ln x})^{'}&=1\\ (e^{\ln x})\ln^{'}x&=1\\ x\ln^{'}x&=1\\ \ln^{'}x&=\frac1x\\ \end{aligned}\]

\[\begin{aligned} (\ln f(x))^{'}=&\ln^{'}f(x)f^{'}(x)\\ =&\frac1{f(x)}f^{'}(x)\\ =&\frac{f^{'}(x)}{f(x)}\\ \end{aligned}\]

幂次法则

\[\begin{aligned} \left(f(x)^{g(x)}\right)^{'}=&\left(\exp(\ln f(x)g(x))\right)^{'}\\ =&\exp(\ln f(x)g(x))(\ln f(x)g(x))^{'}\\ =&f(x)^{g(x)}(\ln f(x)g(x))^{'}\\ =&f(x)^{g(x)}\left(\ln f(x)g^{'}(x)+(\ln f(x))^{'}g(x)\right)\\ =&f(x)^{g(x)}\left(\ln f(x)g^{'}(x)+(\ln f(x))^{'}g(x)\right)\\ =&f(x)^{g(x)}\left(\ln f(x)g^{'}(x)+\frac{f^{'}(x)g(x)}{f(x)}\right)\\ \end{aligned}\]

乘方法则

\[\begin{aligned} (f^c(x))^{'}=&(\exp(c\ln f(x)))^{'}\\ =&\exp(c\ln f(x))(c\ln f(x))^{'}\\ =&cf^c(x)(\ln f(x))^{'}\\ =&cf^c(x)\frac{f^{'}(x)}{f(x)}\\ =&cf^{'}(x)f^{c-1}(x)\\ \end{aligned}\]

一般对数求导

\[\begin{aligned} (\log_a(x))^{'}=&\left(\frac{\ln x}{\ln a}\right)^{'}\\ =&\frac{\ln^{'}x}{\ln a}\\ =&\frac1{x\ln a}\\ (\log_x(a))^{'}=&\left(\frac{\ln a}{\ln x}\right)^{'}\\ =&\ln a({\ln^{-1} x})^{'}\\ =&\ln a\cdot(-1)(\ln^{'} x)({\ln^{-2} x})\\ =&-\ln a\cdot\frac1x({\ln^{-2} x})\\ =&-\frac{\ln a}{x\ln^2x}\\ \end{aligned}\]

除法法则

\[\begin{aligned} \left(\frac{f(x)}{g(x)}\right)^{'}=&\left(f(x)\cdot g^{-1}(x)\right)^{'}\\ =&f^{'}(x)\cdot g^{-1}(x)+f(x)\cdot\left(g^{-1}(x)\right)^{'}\\ =&f^{'}(x)\cdot g^{-1}(x)+f(x)\cdot(-1)g^{'}(x)g^{-2}(x)\\ =&f^{'}(x)\cdot g^{-1}(x)-f(x)\cdot g^{'}(x)g^{-2}(x)\\ =&\frac{f^{'}(x)\cdot g(x)-f(x)\cdot g^{'}(x)}{g^2(x)}\\ \end{aligned}\]

参考

【官方双语】微积分的本质 01-05