Limits and Standard derivatives

2 Properties of limits

• ${lim}_{x\to a}\left[cf\right(x\left)\right]=c{lim}_{x\to a}f\left(x\right)$
• ${lim}_{x\to a}\left[f\right(x)\pm g(x\left)\right]={lim}_{x\to a}f\left(x\right)\pm {lim}_{x\to a}g\left(x\right)$
• ${lim}_{x\to a}\left[f\right(x\left)g\right(x\left)\right]={lim}_{x\to a}f\left(x\right){lim}_{x\to a}g\left(x\right)$
• ${lim}_{x\to a}\frac{f\left(x\right)}{g\left(x\right)}=\frac{{lim}_{x\to a}f\left(x\right)}{{lim}_{x\to a}g\left(x\right)}$ if ${lim}_{x\to a}g\left(x\right)\ne 0$
• ${lim}_{x\to a}\left[f\right(x){]}^n=[{lim}_{x\to a}f\left(x\right){]}^n$
• ${lim}_{x\to a}\left[\sqrt[n]{nf\left(x\right)}\right]=\sqrt[n]{n{lim}_{x\to a}f\left(x\right)}$
• ${lim}_{x\to a}c=c$
• ${lim}_{x\to 0}\sin x=0$
• ${lim}_{x\to 0}\cos x=1$
• ${lim}_{x\to 0}\frac{\sin x}{x}=1$
• ${lim}_{x\to 0}\frac{\tan x}{x}=1$
• ${lim}_{x\to 0}\frac{\log (1+x)}{x}=1$
• ${lim}_{x\to 0}{e}^x=1$
• ${lim}_{x\to 0}\frac{e^x-1}{x}=1$
• ${lim}_{x\to 0}\frac{a^x-1}{x}={\log }_{e}a$
• ${lim}_{x\to \infty }(1+\frac{1}{x}{)}^x=e$
• ${lim}_{x\to 0}(1+x{)}^{1/x}=e$
• ${lim}_{x\to \infty }(1+\frac{a}{x}{)}^x=e^a$
• ${lim}_{x\to \infty }{a}^x=\infty$, when $a>1$
• ${lim}_{x\to \infty }{a}^x=0$, when $a<1$
• ${lim}_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$
• ${lim}_{x\to 0}\frac{(1+x{)}^n-1}{x}=n$
• ${lim}_{n\to \infty }n{x}^n=0$, when $|x|<1$
• ${lim}_{n\to \infty }\frac{x^n}{n!}=0$
• ${lim}_{n\to \infty }\frac{m(m-1)(m-2)\cdots (m-n+1)}{n!}{x}^n=0$, when $|x|<1$

3 Standard derivative rules and formulas

These formulas should be remembered for solving further problems:

• $(cf{)}^{\prime }=c{f}^{\prime }(x)$
• $\frac{d}{dx}\left(c\right)=0$
• $(f\pm g{)}^{\prime }=f^{\prime }(x)\pm g^{\prime }(x)$
• $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$
• $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$ $⟹$ Product rule
• $\frac{d}{dx}\left(f\right(g\left(x\right)\left)\right)=f^{\prime }\left(g\right(x\left)\right)g^{\prime }\left(x\right)$ $⟹$ Chain rule
• $(\frac{f}{g}{)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$ $⟹$ Quotient rule
• $\frac{d}{dx}\left(x\right)=1$
• $\frac{d}{dx}\left(\sin x\right)=\cos x$
• $\frac{d}{dx}\left(\cos x\right)=-\sin x$
• $\frac{d}{dx}\left(\tan x\right)={\sec }^{2}x$
• $\frac{d}{dx}\left(\csc x\right)=-\csc x\cot x$
• $\frac{d}{dx}\left(\sec x\right)=\sec x\tan x$
• $\frac{d}{dx}\left(\cot x\right)=-{\csc }^{2}x$
• $\frac{d}{dx}\left(a^x\right)=a^x\log \left(a\right)$
• $\frac{d}{dx}\left(e^x\right)=e^x$
• $\frac{d}{dx}\left(\ln x\right)=\frac{1}{x},x>0$
• $\frac{d}{dx}\left(\ln |x|\right)=\frac{1}{x},x\ne 0$
• $\frac{d}{dx}\left({\sin }^{-1}x\right)=\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}\left({\cos }^{-1}x\right)=-\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}\left({\tan }^{-1}x\right)=\frac{1}{1+x^2}$
• $\frac{d}{dx}\left({\csc }^{-1}x\right)=-\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}\left({\sec }^{-1}x\right)=\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}\left({\cot }^{-1}x\right)=-\frac{1}{1+x^2}$
• $\frac{d}{dx}\left(\sin \text{h}x\right)=\cos \text{h}x$
• $\frac{d}{dx}\left(\cos \text{h}x\right)=\sin \text{h}x$
• $\frac{d}{dx}\left(\tan \text{h}x\right)=\sec {\text{h}}^{2}x$
• $\frac{d}{dx}\left(\csc \text{h}x\right)=-\csc \text{h}x\cot \text{h}x$
• $\frac{d}{dx}\left(\sec \text{h}x\right)=-\sec \text{h}x\tan \text{h}x$
• $\frac{d}{dx}\left(\cot \text{h}x\right)=-\csc {\text{h}}^{2}x$
• $\frac{d}{dx}\left({\log }_a\right(x\left)\right)=\frac{1}{x\ln a},x>0$

4 Leibnitz’s theorem

Useful for finding $n^{\text{th}}$ derivatives of product functions:

If $y=uv$ where $u$ and $v$ are functions of $x$ possessing ${\text{n}}^{th}$ derivatives, then

$$\begin{array}{cc}{y}_n& ={}^n{C}_0{u}_{n}v+{}^n{C}_1{u}_{n-1}{v}_1+{}^n{C}_2{u}_{n-2}{v}_2+{}^n{C}_3{u}_{n-3}{v}_3+\dots +{}^n{C}_r{u}_{n-r}{v}_r+\dots +{}^n{C}_{n}u{v}_n\\ & =u_{n}v+{}^n{C}_1{u}_{n-1}{v}_1+{}^n{C}_2{u}_{n-2}{v}_2+{}^n{C}_3{u}_{n-3}{v}_3+\dots +{}^n{C}_r{u}_{n-r}{v}_r+\dots +u{v}_n\\ \left(uv{)}_{\left(n\right)}\right(x)& =\sum _{k=0}^n{}^n{C}_k{u}_{(n-k)}\left(x\right)v_{\left(k\right)}\left(x\right),\end{array}$$

where ${}^n{C}_k=\frac{n!}{k!(n-k)!}$ is the binomial coefficient, $u_{\left(0\right)}\left(x\right)=u\left(x\right)$ and $v_{\left(0\right)}\left(x\right)=v\left(x\right)$.