User:Biblefreak123

General

This page is mainly for my testing of posts (or learning the coding sys), for now. As I am relatively new to wiki editing, this seems appropriate to me. If you want to, sign the 'guestbook' section of my talk page... mainly to let me know this is working. Thanks and God Bless!

Sandbox

${\mbox{Thm 1: }}{\frac {d}{dx}}c=0$
${\mbox{Thm 2: }}{\frac {d}{dx}}x^{n}=nx^{n-1}$
${\mbox{Thm 3: }}{\frac {d}{dx}}cf(x)=c{\frac {d}{dx}}f(x)$
${\mbox{Thm 4: }}{\frac {d}{dx}}(f(x)\pm g(x))={\frac {d}{dx}}f(x)\pm {\frac {d}{dx}}g(x)$
${\mbox{Thm 5: }}{\frac {d}{dx}}(f(x)g(x))=f'(x)g(x)+f(x)g'(x)$
${\mbox{Thm 6: }}{\frac {d}{dx}}{\frac {f(x)}{g(x)}}={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}$
${\mbox{Thm 7: }}{\frac {d}{dx}}sin(x)=cos(x)$
${\mbox{Thm 8: }}{\frac {d}{dx}}cos(x)=-sin(x)$
${\mbox{Thm 9: }}{\frac {d}{dx}}tan(x)=sec^{2}(x)$
${\mbox{Thm 10: }}{\frac {d}{dx}}cot(x)=-csc^{2}(x)$
${\mbox{Thm 11: }}{\frac {d}{dx}}sec(x)=sec(x)tan(x)$
${\mbox{Thm 12: }}{\frac {d}{dx}}csc(x)=-csc(x)cot(x)$
${\mbox{Thm 13: }}{\frac {d}{dx}}f(g(x))=f'(g(x))g'(x)$
${\mbox{Thm 14: }}{\frac {d}{dx}}e^{x}=e^{x}$
${\mbox{Thm 15: }}{\frac {d}{dx}}ln(x)={\frac {1}{x}}\quad {\frac {d}{dx}}ln|x|={\frac {1}{x}}$
${\mbox{Thm 16: }}{\frac {d}{dx}}log_{b}(x)={\frac {1}{x\,ln(b)}}$
${\mbox{Thm 17: }}{\frac {d}{dx}}a^{x}=a^{x}ln(a)$
${\mbox{Thm 18: }}{\frac {d}{dx}}sin^{-1}(x)={\frac {1}{\sqrt {1-x^{2}}}}$
${\mbox{Thm 19: }}{\frac {d}{dx}}cos^{-1}(x)={\frac {-1}{\sqrt {1-x^{2}}}}$
${\mbox{Thm 20: }}{\frac {d}{dx}}tan^{-1}(x)={\frac {1}{1+x^{2}}}$
${\mbox{Thm 21: }}{\frac {d}{dx}}cot^{-1}(x)={\frac {-1}{1+x^{2}}}$
${\mbox{Thm 22: }}{\frac {d}{dx}}sec^{-1}(x)={\frac {1}{x{\sqrt {x^{2}-1}}}}$
${\mbox{Thm 23: }}{\frac {d}{dx}}sec^{-1}(x)={\frac {-1}{x{\sqrt {x^{2}-1}}}}$


${\frac {d}{dx}}y={\frac {dy}{dx}}$
${\frac {d}{dx}}y^{2}={\frac {d}{dx}}f(g(x)){\big [}{\mbox{ where }}f(x)=x^{2}{\mbox{ and }}g(x)=y{\big ]}=2y{\frac {dy}{dx}}$


${\frac {d}{dx}}{\bigg [}x^{3}+y^{3}=6xy{\bigg ]}$
$3x^{2}+3y^{2}{\frac {dy}{dx}}=6(y+{\frac {dy}{dx}}x)$
$3y^{2}{\frac {dy}{dx}}-6x{\frac {dy}{dx}}=6y-3x^{2}$
${\frac {dy}{dx}}={\frac {6y-3x^{2}}{3y^{2}-6x}}$



$y=cosx$
$y'=-sinx$
$y''=-cosx$
$y'''=sinx$
$y^{(4)}=cosx$
${\frac {d^{0}}{dx^{0}}}cosx={\frac {d^{4}}{dx^{4}}}cosx=cosx$
${\frac {d^{27}}{dx^{27}}}cosx={\frac {d^{4*6+3}}{dx^{4*6+3}}}cosx={\frac {d^{3}}{dx^{3}}}cosx=sinx$
${\frac {d}{dx}}{\bigg [}y=x^{tan(x)}{\bigg ]}$
${\frac {d}{dx}}{\bigg [}ln(y)=ln(x^{tan(x)}){\bigg ]}$
${\frac {d}{dx}}{\bigg [}ln(y)=tan(x)ln(x){\bigg ]}$
${\frac {1}{y}}{\frac {dy}{dx}}=sec^{2}(x)ln(x)+{\frac {tan(x)}{x}}$
${\frac {dy}{dx}}=x^{tan(x)}{\bigg [}sec^{2}(x)ln(x)+{\frac {tan(x)}{x}}{\bigg ]}$

$sinh(x)={\frac {e^{x}-e^{-x}}{2}}$
$cosh(x)={\frac {e^{x}+e^{-x}}{2}}$
$tanh(x)={\frac {sinh(x)}{cosh(x)}}$
$csch(x)={\frac {1}{sinh(x)}}$
$sech(x)={\frac {1}{cosh(x)}}$
$coth(x)={\frac {cosh(x)}{sinh(x)}}$

$sinh^{-1}(x)=ln(x+{\sqrt {x^{2}+1}})x\epsilon R$
$cosh^{-1}(x)=ln(x+{\sqrt {x^{2}-1}})x\geq 1$
$tanh^{-1}(x)={\frac {1}{2}}ln({\frac {1+x}{1-x}})-1<x<1$

${\frac {d}{dx}}sinh^{-1}(x)={\frac {1}{\sqrt {1+x^{2}}}}$
${\frac {d}{dx}}cosh^{-1}(x)={\frac {1}{\sqrt {x^{2}-1}}}$
${\frac {d}{dx}}tanh^{-1}(x)={\frac {1}{1-x^{2}}}$
${\frac {d}{dx}}csch^{-1}(x)=-{\frac {1}{|x|{\sqrt {x^{2}+1}}}}$
${\frac {d}{dx}}sech^{-1}(x)=-{\frac {1}{x{\sqrt {1-x^{2}}}}}$
${\frac {d}{dx}}coth^{-1}(x)={\frac {1}{1-x^{2}}}$