Calculus

Profile · 07:36 PM

$\reverse \begin{tabular}{|c|c|} \hline\\ f(x)&f'(x)\\ \hline\\ sin x&cos x\\ tan x&sec^2 x\\ sec x&sec x tan x\\ sin^{-1} x&\frac{1}{\sqrt{1-x^2}}\\ tan^{-1} x&\frac{1}{x^2+1}\\ sec^{-1} x&\frac{1}{|x|\sqrt{x^2-1}}\\ ln x&\frac{1}{x}\\ sinh x&cosh x\\ cosh x&sinh x\\ tanh x&sech^2 x\\ coth x&-csch^2 x\\ sech x&-sech x tanh x\\ scsh x&-csch x coth x\\ sinh^{-1} x&\frac{1}{\sqrt{1+x^2}}\\ cosh^{-1} x&\frac{1}{\sqrt{x^2-1}}, x>1\\ tanh^{-1} x&\frac{1}{1-x^2},|x|<1\\ coth^{-1} x&\frac{1}{1-x^2},|x|>1\\ sech^{-1} x&-\frac{1}{x\sqrt{1-x^2}},0<x<1\\ csch^{-1} x&-\frac{1}{|x|\sqrt{1+x^2}},x\neq0\\ \hline \end{tabular}$
$\reverse \begin{tabular}{c} sinh x=\frac{e^x-e^{-x}}{2}\\ cosh x=\frac{e^x+e^{-x}}{2}\\ sin x=\frac{e^{ix}-e^{-ix}}{2i}\\ cos x=\frac{e^{ix}+e^{-ix}}{2}\\ \end{tabular}$
$\reverse \begin{tabular}{c} sin (-x)=-sin x\\ cos (-x)=cos x\\ sin^2 x+cos^2 x=1\\ sec^2 x=1+ tan^2 x\\ csc^2 x=1+ cot^2 x\\ sin (a+b)=sin a cos b+cos a sin b\\ cos (a+b)=cos a cos b - sin a sin b\\ cos^2 x=\frac{1+cos 2x}{2}\\ sin^2 x=\frac{1-cos 2x}{2}\\ cosh^2 x - sinh^2 x =1\\ sinh 2x=2 sinh x cosh x\\ cosh 2x=cosh^2 x + sinh^2 x\\ cosh^2 x=\frac{cosh 2x +1}{2}\\ sinh^2 x=\frac{cosh 2x -1}{2}\\ tanh^2 x = 1 - sech^2 x\\ coth^2 x=1+csch^2 x\\ \end{tabular}$
$\reverse \begin{tabular}{l} \lim_{n\to\infty}\frac{ln n}{n}=0\\ \lim_{n\to\infty}n^{\frac{1}{n}}=1\\ \lim_{n\to\infty}x^{\frac{1}{n}}=1,x>0\\ \lim_{n\to\infty}(1+\frac{x}{n})^n=e^x\\ \lim_{n\to\infty}x^n=0,|x|<1\\ \lim_{n\to\infty}\frac{x^n}{n!}=0\\ \end{/tabular}$
arc length $\reverse \int\sqrt{1+(f')^2}$
surface area $\reverse \int 2\pi f\sqrt{1+(f')^2}$

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