Calculus

tags math calculus

Differential

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)+f(x)}{h}$$ $$f^{\prime }(x)=\frac{d}{dx}f(x)$$ $$\frac{d}{dx}f(x)g(x)=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$$ $$\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{f^{\prime }(x)g(x)+f(x)g^{\prime }(x)}{g(x{)}^2}$$ $$\frac{d}{dx}f(g(x))=f^{\prime }(g(x))\times g^{\prime }(x)$$

example

$$\frac{d}{dx}cos(x)=-sin(x)$$ $$\frac{d}{dx}tan(x)=sec(x{)}^2$$ $$\frac{d}{dx}cot(x)=csc(x{)}^2$$ $$\frac{d}{dx}si{n}^{-1}(x)=\frac{1}{\sqrt{1-x^2}}$$ $$\frac{d}{dx}co{s}^{-1}(x)=\frac{-1}{\sqrt{1-x^2}}$$ $$\frac{d}{dx}ta{n}^{-1}(x)=\frac{1}{1+x^2}$$

intergrant

$$\begin{gathered} g(x)=\underset{{}_{k(x)}}{{\int }^{h(x)}}f(x)dx \\ {g}^{\prime }(x)=f(h(x))h^{\prime }(x)-f(k(x))k^{\prime }(x) \end{gathered}$$ $$\int f(x)dg(x)=f(x)g(x)-\int g(x)df(x)$$

example

$$\int e^{f(x)}dx=\frac{e^{f(x)}}{f^{\prime }(x)}+c$$ $$\int \ln (x)dx=x\ln (x)-x+c$$ $$\int \frac{1}{x}dx=\ln (x)+c$$ $$\int sin(x)dx=-cos(x)+c$$ $$\int tan(x)dx=-\ln (cos(x))+c$$ $$\int sec(x{)}^{2}dx=tan(x)+c$$ $$\int \frac{2x}{1+x^2}dx=\ln (1+x^2)+c$$

Taylor series 泰勒展開式

$$\begin{gathered} f^{(n)}(a)=f(x)在x=a的n次導數 \\ f(x)=\frac{f^{(0)}(a)}{0!}(x-a{)}^0+\frac{f^{(1)}(a)}{1!}(x-a{)}^1+\frac{f^{(2)}(a)}{2!}(x-a{)}^2\cdots \\ f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n \end{gathered}$$

find sin(x) Taylor series

$$\begin{gathered} \begin{array}{rl}f(x)=sin(x)\to & f^0(x)=sin(x)\\ & f^1(x)=cos(x)\\ & f^2(x)=-sin(x)\\ & f^3(x)=-cos(x)\\ & f^4(x)=sin(x)=f^0(x)\end{array} \\ \text{if }a=0 \\ \begin{array}{rl}f(x)& =\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n\\ & =\sum _{n=0}^{\infty }\frac{sin(0)}{(4n+0)!}(x{)}^{4n}+\frac{cos(0)}{(4n+1)!}(x{)}^{4n+1}+\frac{-sin(0)}{(4n+2)!}(x{)}^{4n+2}+\frac{-cos(0)}{(4n+3)!}(x{)}^{4n+3}\\ & =\sum _{n=0}^{\infty }\frac{cos(0)}{(4n+1)!}(x{)}^{4n+1}+\frac{-cos(0)}{(4n+3)!}(x{)}^{4n+3}\\ & =\sum _{n=0}^{\infty }\frac{1}{(4n+1)!}(x{)}^{4n+1}+\frac{-1}{(4n+3)!}(x{)}^{4n+3}\\ sin(x)& =\sum _{n=0}^{\infty }\frac{(-1{)}^n}{(2n+1)!}(x{)}^{2n+1}\end{array} \end{gathered}$$