Lys van afgeleides

Saam met integrasie vorm differensiasie die hoofbewerkings van kalkulus. In die onderstaande lys is f en g differensieerbare funksies van die reële getal s. c is ook 'n reële getal.

Hierdie lys van afgeleides is voldoende om enige elementêre funksie te differensieer.

Algemene reëls by die afleiding van funksies

$\left(cf\right)=c{f}^{\prime }$

$\left(f+g\right)=f^{\prime }+g^{\prime }$

Produkreël

$\left(fg\right)=f^{\prime }g+f{g}^{\prime }$

Kwosiëntreël

$\left(\frac{f}{g}\right)=\frac{f^{\prime }g-f{g}^{\prime }}{g^2},g\ne 0$

Kettingreël

$(f\circ g{)}^{\prime }=(f^{\prime }\circ g)g^{\prime }$

Afgeleides van eenvoudige funksies

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

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

$\frac{d}{dx}cx=c$

$\frac{d}{dx}|x|=\frac{|x|}{x}=\mathrm{sgn}x,x\ne 0$

$\frac{d}{dx}{x}^c=c{x}^{c-1}\text{met beide }{x}^c\text{ en }c{x}^{c-1}\text{ gedefinieer}$

$\frac{d}{dx}\left(\frac{1}{x}\right)=\frac{d}{dx}\left(x^{-1}\right)=-x^{-2}=-\frac{1}{x^2}$

$\frac{d}{dx}\left(\frac{1}{x^c}\right)=\frac{d}{dx}\left(x^{-c}\right)=-\frac{c}{x^{c+1}}$

$\frac{d}{dx}\sqrt{x}=\frac{d}{dx}{x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}},x>0$

Afgeleides van eksponensiële funksie en logaritmes

$\frac{d}{dx}{c}^x=c^x\ln c,c>0$

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

$\frac{d}{dx}{\log }_{c}x=\frac{1}{x\ln c},c>0,c\ne 1$

$\frac{d}{dx}\ln x=\frac{1}{x},x>0$

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

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

Afgeleides van trigonometriese funksies

$\frac{d}{dx}\sin x=\cos x$ $\frac{d}{dx}\cos x=-\sin x$ $\frac{d}{dx}\tan x={\sec }^{2}x=\frac{1}{{\cos }^{2}x}$ $\frac{d}{dx}\sec x=\tan x\sec x$ $\frac{d}{dx}\cot x=-{\csc }^{2}x=\frac{-1}{{\sin }^{2}x}$ $\frac{d}{dx}\csc x=-\csc x\cot x$

$\frac{d}{dx}\text{bgsin}x=\frac{1}{\sqrt{1-x^2}}$ $\frac{d}{dx}\text{bgcos}x=\frac{-1}{\sqrt{1-x^2}}$ $\frac{d}{dx}\text{bgtan}x=\frac{1}{1+x^2}$ $\frac{d}{dx}\text{bgsec}x=\frac{1}{|x|\sqrt{x^2-1}}$ $\frac{d}{dx}\text{bgcot}x=\frac{-1}{1+x^2}$ $\frac{d}{dx}\text{bgcsc}x=\frac{-1}{|x|\sqrt{x^2-1}}$

Afgeleides van hiperboliese funksies

$\frac{d}{dx}\sinh x=\cosh x=\frac{e^x+e^{-x}}{2}$ $\frac{d}{dx}\cosh x=\sinh x=\frac{e^x-e^{-x}}{2}$ $\frac{d}{dx}\tanh x={\mathrm{sech}}^{2}x$ $\frac{d}{dx}\mathrm{sech}x=-\tanh x\mathrm{sech}x$ $\frac{d}{dx}\coth x=-{\mathrm{csch}}^{2}x$ $\frac{d}{dx}\mathrm{csch}x=-\coth x\mathrm{csch}x$

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

Afgeleides van inverse funksies

$\frac{d}{dx}(f^{-1}(x))=\frac{1}{f^{\prime }(f^{-1}(x))}$

Verwysings

1. Stewart, J. (2003). Single Variable Calculus. (5th ed.). Belmont, USA: Thomson Learning.
2. Groenewald, G.J., Hitge, M. (2005). Analise II Studiegids vir WISK121A. Potchefstroom: Noordwes-Universiteit.
3. Jordan, D.W., Smith, P. (2002). Mathematical techniques: An introduction for the engineering, physical and mathematical sciences. USA: Oxford University Press.

Aantekeninge

• Sien ook Lys van integrale

• Hierdie artikel is 'n vertaling van die Engelse Wikipedia artikel Table of derivatives