Up: [[수학,math]]

Sub:
 [[역함수의_도함수(미분)_증명]]
 확률:
  [[여집합의_확률_증명]]
 미분:
  [[sin_x_미분_증명]] $(\sin x)'=\cos x$
  [[cos_x_미분_증명]] $(\cos x)'=-\sin x$
  [[tan_x_미분_증명]] $(\tan x)'=\sec^2 x$
  [[csc_x_미분_증명]] $(\csc x)'=-\csc x \cot x$
  [[sec_x_미분_증명]] $(\sec x)'=\sec x \tan x$
  [[cot_x_미분_증명]] $(\cot x)'=-\csc^2 x$
  [[arcsin_x(아크사인)_미분_증명]] $(\sin^{-1}x)'=\frac1{\sqrt{1-x^2}}$
  [[arccos_x(아크코사인)_미분_증명]]
  [[arctan_x(아크탄젠트)_미분_증명]]
  [[arccsc_x(아크코시컨트)_미분_증명]]
  [[arcsec_x(아크시컨트)_미분_증명]]
  [[arccot_x(아크코탄젠트)_미분_증명]]
  [[sinh_x_미분_증명]] $(\sinh x)'=\cosh x$
  [[cosh_x_미분_증명]] $(\cosh x)'=\sinh x$
  [[tanh_x_미분_증명]] $(\tanh x)'=\operatorname{sech}^2 x$
   todo add $(\tanh x)' = 1 - \tanh^2 x$
  [[csch_x_미분_증명]] $(\operatorname{csch} x)'=-\operatorname{csch}x\operatorname{coth}x$
{
$(\operatorname{csch}x)'$
 $=\left(\frac1{\sinh x}\right)'$
 $=\frac{-\cosh x}{\sinh^2 x}$
 $=\frac{-1}{\sinh x}\cdot\frac{\cosh x}{\sinh x}$
 $=-\operatorname{csch}x\operatorname{coth}x$
}
  [[sech_x_미분_증명]] $({\rm sech}x)'=-\operatorname{sech}x \operatorname{tanh}x$
{
$(\operatorname{sech}x)'$
 $=\left(\frac1{\operatorname{cosh}x}\right)'$
 $=\frac{-\sinh x}{\cosh^2 x}$
 $=\frac{-1}{\cosh x}\cdot \frac{\sinh x}{\cosh x}$
 $=-\operatorname{sech}x \operatorname{tanh}x$
}
  [[coth_x_미분_증명]] $({\rm coth}x)'=$
  [[VG:역쌍곡선함수,inverse_hyperbolic_function]]의 미분:
  [[arcsinh_x_미분_증명]] ${d\over dx}(\sinh^{-1}x)=\frac1{\sqrt{1+x^2}}$
  [[arccosh_x_미분_증명]] ${d\over dx}(\cosh^{-1}x)=\frac1{\sqrt{x^2-1}}$
  [[arctanh_x_미분_증명]] ${d\over dx}(\tanh^{-1}x)=\frac1{1-x^2}$
  ...
 적분:
  [[sin_x_적분_증명]]
  [[cos_x_적분_증명]]
  [[tan_x_적분_증명]] $\textstyle\int\tan x dx=-\ln|\cos x|+C$
  [[csc_x_적분_증명]] $\textstyle\int\csc xdx=-\ln|\csc x+\cot x|+C$
  [[sec_x_적분_증명]] $\textstyle\int\sec xdx=\ln|\sec x+\tan x|+C$
  [[cot_x_적분_증명]]
  ..
  [[arcsinh_x_적분_증명]]
  [[arccosh_x_적분_증명]]
  [[arctanh_x_적분_증명]]
  [[arccsch_x_적분_증명]]
  [[arcsech_x_적분_증명]]
  [[arccoth_x_적분_증명]]


[[루트엑스적분법,integral_root_x_dx]]

[[arcsinh_증명]] $\sinh^{-1}x=\ln(x+\sqrt{x^2+1})$
[[arccosh_증명]] $\cosh^{-1}x=\ln\left(x+\sqrt{x^2-1}\right)$
[[arctanh_증명]] $\tanh^{-1}x=\frac12\ln\left(\frac{1+x}{1-x}\right)$
----
See also [[삼각함수적분표,trigonometric_integrals]]