Kaidah Penurunan Umum

Kaidah Penurunan Umum

Kaidah Penurunan Umum:

1. Kaidah Kelinieran

2. Kaidah Darab



3. Kaidah Hasil-Bagi



4. Kaidah Rantai



5. Turunan Fungsi Invers


untuk setiap fungsi terdiferensialkan f dengan argumen riil dan dengan nilai riil, bila komposisi dan invers ada.

6. Kaidah Pangkat Umum



Turunan Fungsi Sederhana

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Turunan Fungsi Eksponensial dan Logaritmik

Perhatikan bahwa persamaan tersebut berlaku untuk semua c, namun turunan tersebut menghasilkan bilangan kompleks

Turunan Fungsi Trigonometrik

${\displaystyle (\sin x)'=\cos x\,}$	${\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,}$
${\displaystyle (\cos x)'=-\sin x\,}$	${\displaystyle (\arccos x)'={-1 \over {\sqrt {1-x^{2}}}}\,}$
${\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}\,}$	${\displaystyle (\arctan x)'={1 \over 1+x^{2}}\,}$
${\displaystyle (\sec x)'=\sec x\tan x\,}$	${\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1}}}\,}$
${\displaystyle (\csc x)'=-\csc x\cot x\,}$	${\displaystyle (\operatorname {arccsc} x)'={-1 \over |x|{\sqrt {x^{2}-1}}}\,}$
${\displaystyle (\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}\,}$	${\displaystyle (\operatorname {arccot} x)'={-1 \over 1+x^{2}}\,}$

Turunan Fungsi Hiperbolik

${\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}$	${\displaystyle (\operatorname {arcsinh} \,x)'={1 \over {\sqrt {x^{2}+1}}}}$
${\displaystyle (\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}}$	${\displaystyle (\operatorname {arccosh} \,x)'={1 \over {\sqrt {x^{2}-1}}}}$
${\displaystyle (\tanh x)'=\operatorname {sech} ^{2}\,x}$	${\displaystyle (\operatorname {arctanh} \,x)'={1 \over 1-x^{2}}}$
${\displaystyle (\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x}$	${\displaystyle (\operatorname {arcsech} \,x)'={-1 \over x{\sqrt {1-x^{2}}}}}$
${\displaystyle (\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}$	${\displaystyle (\operatorname {arccsch} \,x)'={-1 \over x{\sqrt {1+x^{2}}}}}$
${\displaystyle (\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x}$	$$

{\displaystyle (\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}}	{\displaystyle (\operatorname {arcsinh} \,x)'={1 \over {\sqrt {x^{2}+1}}}}
{\displaystyle (\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}}	{\displaystyle (\operatorname {arccosh} \,x)'={1 \over {\sqrt {x^{2}-1}}}}
{\displaystyle (\tanh x)'=\operatorname {sech} ^{2}\,x}	{\displaystyle (\operatorname {arctanh} \,x)'={1 \over 1-x^{2}}}
{\displaystyle (\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x}	{\displaystyle (\operatorname {arcsech} \,x)'={-1 \over x{\sqrt {1-x^{2}}}}}
{\displaystyle (\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x}	{\displaystyle (\operatorname {arccsch} \,x)'={-1 \over x{\sqrt {1+x^{2}}}}}
{\displaystyle (\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x}

Turunan Fungsi Khusus

{\displaystyle (\Gamma (x))'=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt} {\displaystyle (\Gamma (x))'=\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)=\Gamma (x)\psi (x)} Fungsi gamma ${\displaystyle (\Gamma (x))'=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt}$ ${\displaystyle (\Gamma (x))'=\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)=\Gamma (x)\psi (x)}$ Fungsi Riemann Zeta ${\displaystyle (\zeta (x))'=-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots \!}$ ${\displaystyle (\zeta (x))'=-\sum _{p{\text{ prime}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ prime}},q\neq p}{\frac {1}{1-q^{-x}}}\!}$

{\displaystyle (\zeta (x))'=-\sum _{p{\text{ prime}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ prime}},q\neq p}{\frac {1}{1-q^{-x}}}\!}