미분법

1 곱미분

$$\begin{array}{rl}F(x)& =f(x)g(x)\\ F^{\prime }(x)& =\underset{h\to 0}{\lim }\frac{F(x+h)-F(x)}{h}\\ & =\underset{h\to 0}{\lim }\frac{f(x+h)g(x+h)-f(x)g(x)}{h}\\ & =\underset{h\to 0}{\lim }\frac{f(x+h)g(x+h)-f(x+h)g(x)+f(x+h)g(x)-f(x)g(x)}{h}\\ & =\underset{h\to 0}{\lim }f(x+h)\underset{h\to 0}{\lim }\frac{g(x+h)-g(x)}{h}+g(x)\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}\\ & =f^{\prime }(x)g(x)+f(x)g^{\prime }(x)\end{array}$$

2 몫미분

$$\begin{array}{r}F(x)=\frac{f(x)}{g(x)}(g(x)\ne 0)\end{array}$$ $$\begin{array}{rl}f(x)& =F(x)g(x)\\ & =F(x)g^{\prime }(x)+F^{\prime }(x)g(x)\end{array}$$ $$\begin{array}{rl}{F}^{\prime }(x)& =\frac{f^{\prime }(x)-F(x)g^{\prime }(x)}{g(x)}\\ & =\frac{f^{\prime }(x)-\frac{f(x)}{g(x)}{g}^{\prime }(x)}{g(x)}\\ & =\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}\end{array}$$ $$\therefore F^{\prime }(x)=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$$

3 역함수 미분

$$\begin{array}{r}{f}^{-1}(x)=g(x)\\ f(g(x))=x\\ f^{\prime }(g(x))\cdot g^{\prime }(x)=1\\ g^{\prime }(x)=\frac{1}{f^{\prime }(g(x))}\end{array}\text{\hspace{1em}(1)}$$ $$\begin{array}{r}f(y)=x\\ f^{\prime }(y)\frac{dy}{dx}=1\\ \frac{dx}{dy}=\frac{1}{f^{\prime }(y)}\end{array}\text{\hspace{1em}(2)}$$

$\frac{dx}{dy}=g^{\prime }(x),y=g(x)$이므로

$$\begin{array}{r}{g}^{\prime }(x)=\frac{1}{f^{\prime }(g(x))}\end{array}$$