חשבון אינפיניטסימלי/טבלת דיפרנציאלים

חוקים כללים

${\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}f(x)\pm {\frac {d}{dx}}g(x)$

${\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}$

${\frac {d}{dx}}[f(x)\cdot g(x)]=f(x)\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}+g(x)\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}$

${\frac {d}{dx}}\left[{\frac {f(x)}{g(x)}}\right]={\frac {g(x)\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}-f(x)\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}}{g(x)^{2}}}$

${\frac {d}{dx}}\left[f(g(x))\right]=f'(g(x))\cdot g'(x)$

חזקות ופולינומים

${\frac {d}{dx}}(c)=0$

${\frac {d}{dx}}x=1$

${\frac {d}{dx}}x^{n}=nx^{n-1}$

${\frac {d}{dx}}{\sqrt {x}}={\frac {1}{2{\sqrt {x}}}}$

${\frac {d}{dx}}{\frac {1}{x}}=-{\frac {1}{x^{2}}}$

הפונקציות הטריגונומטריות

${\frac {d}{dx}}\sin(x)=\cos(x)$

${\frac {d}{dx}}\cos(x)=-\sin(x)$

${\frac {d}{dx}}\tan(x)=\sec ^{2}(x)$

${\frac {d}{dx}}\cot(x)=-\csc ^{2}(x)$

${\frac {d}{dx}}\sec(x)=\sec(x)\cdot \tan(x)$

${\frac {d}{dx}}\csc(x)=-\csc(x)\cdot \cot(x)$

הפונקציות הטריגונומטריות ההפוכות

${\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}}$

${\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}}$

${\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}$

${\frac {d}{dx}}\operatorname {arcsec}(x)={\frac {1}{|x|{\sqrt {x^{2}-1}}}}$

${\frac {d}{dx}}\operatorname {arccot}(x)=-{\frac {1}{1+x^{2}}}$

${\frac {d}{dx}}\operatorname {arccsc}(x)=-{\frac {1}{|x|{\sqrt {x^{2}-1}}}}$

הפונקציות ההיפרבוליות

${\frac {d}{dx}}\sinh(x)=\cosh(x)$

${\frac {d}{dx}}\cosh(x)=\sinh(x)$

${\frac {d}{dx}}\tanh(x)={\mbox{sech}}^{2}(x)$

${\frac {d}{dx}}{\mbox{sech}}(x)=-\tanh(x)\cdot {\mbox{sech}}(x)$

${\frac {d}{dx}}{\mbox{coth}}(x)=-{\mbox{csch}}^{2}(x)$

${\frac {d}{dx}}{\mbox{csch}}(x)=-{\mbox{coth}}(x)\cdot {\mbox{csch}}(x)$

הפונקציות ההיפרבוליות ההפוכות

${\frac {d}{dx}}{\mbox{arsinh}}(x)={\frac {1}{\sqrt {x^{2}+1}}}$

${\frac {d}{dx}}{\mbox{arcosh}}(x)=-{\frac {1}{\sqrt {x^{2}-1}}}$

${\frac {d}{dx}}{\mbox{artanh}}(x)={\frac {1}{1-x^{2}}}$

${\frac {d}{dx}}{\mbox{arsech}}(x)={\frac {1}{x{\sqrt {1-x^{2}}}}}$

${\frac {d}{dx}}{\mbox{arcoth}}(x)={\frac {1}{1-x^{2}}}$

${\frac {d}{dx}}{\mbox{arcsch}}(x)=-{\frac {1}{|x|{\sqrt {x^{2}+1}}}}$

הפונקציה המעריכית ולוגרתמים

${\frac {d}{dx}}e^{x}=e^{x}$

${\frac {d}{dx}}a^{x}=a^{x}\cdot \ln(a)\qquad {\mbox{if }}a>0$

${\frac {d}{dx}}\ln(|x|)={\frac {1}{x}}$

${\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}}\qquad {\mbox{if }}a>0,a\neq 1$

${\frac {d}{dx}}[f(x)^{g(x)}]={\frac {d}{dx}}\left[e^{g(x)\cdot \ln[f(x)]}\right]=f(x)^{g(x)}\cdot \left({\frac {g(x)}{f(x)}}\cdot {\bigg [}{\frac {d}{dx}}f(x){\bigg ]}+\ln[f(x)]\cdot {\bigg [}{\frac {d}{dx}}g(x){\bigg ]}\right),\qquad f>0$

${\frac {d}{dx}}c^{f(x)}={\frac {d}{dx}}\left(e^{f(x)\cdot \ln(c)}\right)=c^{f(x)}\cdot \ln(c)\cdot \left[{\frac {d}{dx}}f(x)\right]$