29  Calculus

29.1 Fundamental limits

$\underset{x\to 0}{lim}\frac{\sin x}{x}=1$	$\underset{x\to 0}{lim}\frac{\ln (1+x)}{x}=1$	$\underset{x\to 0}{lim}\frac{e^x-1}{x}=1$
$\underset{x\to 0}{lim}{(1+\frac{1}{x})}^x=e$	$\underset{x\to 0}{lim}\frac{{\log }_a(1+x)}{x}={\log }_{a}e$	$\underset{x\to 0}{lim}\frac{a^x-1}{x}=\ln a$

Table 29.1: Fundamental limits

29.2 Derivatives

Function $f(x)$	Derivative $f^{\prime }(x)$
$y=a,a\in \mathbb{R}$	$y^{\prime }=0$
$y=x^n,n\in \mathbb{N}$	$y^{\prime }=n{x}^{n-1}$
$y=x^{\alpha },\alpha \in \mathbb{R}$	$y^{\prime }=\alpha x^{\alpha -1}$
$y=x^{\frac{1}{n}},n>0$	$y^{\prime }=\frac{1}{n}{x}^{\frac{1}{n}-1}$
$y=\sin x$	$y^{\prime }=\cos x$
$y=\cos x$	$y^{\prime }=-\sin x$
$y=\tan x$	$y^{\prime }=\frac{1}{{\cos }^{2}x}=1+{\tan }^{2}x$
$y=\cot x$	$y^{\prime }=-\frac{1}{{\sin }^{2}x}=-(1+{\cot }^{2}x)$
$y=\arcsin x$	$y^{\prime }=\frac{1}{\sqrt{1-x^2}}$
$y=\arccos x$	$y^{\prime }=-\frac{1}{\sqrt{1-x^2}}$
$y=\arctan x$	$y^{\prime }=\frac{1}{1+x^2}$
$y=\text{arccot}x$	$y^{\prime }=-\frac{1}{1+x^2}$
$y=a^x$	$y^{\prime }=a^x\ln a$
$y=e^x$	$y^{\prime }=e^x$
$y={\log }_{a}x$	$y^{\prime }=\frac{1}{x\ln a}$
$y=\ln x$	$y^{\prime }=\frac{1}{x}$
$f(x)=c\cdot g(x)$	$f^{\prime }(x)=c\cdot g^{\prime }(x)$
$f(x)=g(x)+s(x)$	$f^{\prime }(x)=g^{\prime }(x)+s^{\prime }(x)$
$f(x)=\frac{1}{g(x)}$	$f^{\prime }(x)=-\frac{g^{\prime }(x)}{g(x{)}^2}$

Table 29.2: Fundamental derivatives

• Derivative of the product: $$[f(x)\cdot g(x){]}^{\prime }(x)=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)\text{.}$$

• Derivative of the ratio: $${\left[\frac{f(x)}{g(x)}\right]}^{\prime }(x)=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}\text{.}$$

• Derivative of the composition: $$[f(g(x)){]}^{\prime }(x)=g^{\prime }(x)\cdot f^{\prime }(g(x))\text{.}$$

Example derivative of the composition

For example $f(g(x))=\ln (1+2x)$, then $f(x)=\ln (x)$ and $g(x)=1+2x$, hence $$[\ln (1+2x){]}^{\prime }(x)=[1+2x{]}^{\prime }(x)\cdot [\ln (x){]}^{\prime }(1+x)\text{.}$$

29.2.1 Taylor series

$$\begin{array}{}\text{(29.1)}& f(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{f^{(n)}(a)}{n!}(x-a{)}^n=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\dots \end{array}$$

29.3 Integrals

Immediate	General
$\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$	$\int f(x{)}^{n}dx=\frac{f(x{)}^{n+1}}{n+1}+c$
$\int \frac{1}{x}dx=\log (x)+c$	$\int \frac{f^{\prime }(x)}{f(x)}dx=\log (f(x))+c$
$\int a^{x}dx={\log }_a(e)+c$	$\int a^{f(x)}{f}^{\prime }(x)dx=a^{f(x)}{\log }_a(e)+c$
$\int e^{x}dx=e^x+c$	$\int e^x{f}^{\prime }(x)dx=e^{f(x)}+c$
$\int \sin (x)dx=-\cos (x)+c$	$\int \sin (f(x))f^{\prime }(x)dx=-\cos (f(x))+c$
$\int \cos (x)dx=\sin (x)+c$	$\int \cos (f(x))f^{\prime }(x)dx=\sin (f(x))+c$
$\int \frac{1}{\cos (x{)}^2}dx=\tan (x)+c$	$\int \frac{f^{\prime }(x)}{\cos (f(x){)}^2}dx=\tan (f(x))+c$
$\int \frac{1}{\sin (x{)}^2}dx=\cot (x)+c$	$\int \frac{f^{\prime }(x)}{\sin (f(x){)}^2}dx=\cot (f(x))+c$
$\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin (x)+c$	$\int \frac{f^{\prime }(x)}{\sqrt{1-f(x{)}^2}}dx=\arcsin (f(x))+c$
$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin (\frac{x}{a})+c$	$\int \frac{f^{\prime }(x)}{\sqrt{a^2-f(x{)}^2}}dx=\arcsin (\frac{f(x)}{a})+c$
$\int \frac{1}{\sqrt{1+x^2}}dx=\arctan (x)+c$	$\int \frac{f^{\prime }(x)}{\sqrt{1+f(x{)}^2}}dx=\arctan (f(x))+c$

Table 29.3: Fundamental integrals

29.3.1 Fundamental theorem

$$f(b)-f(a)={\int }_a^b{f}^{\prime }(x)dx⟺\int f^{\prime }(x)dx=f(x)+C\text{.}$$

29.3.2 Integration by parts

$${\int }_a^{b}f(x)g^{\prime }(x)dx={\left[f(x)g(x)\right]}_{x=a}^{x=b}-{\int }_a^b{f}^{\prime }(x)g(x)dx\text{,}$$ or in compact form: $${\int }_a^{b}f(x)dg(x)={\left[f(x)g(x)\right]}_{x=a}^{x=b}-{\int }_a^{b}g(x)df(x)dx\text{.}$$

29.4 Useful relations

• Newton binomial

$$(a+b{)}^n=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k\text{.}$$