Spisak diferencijacijskih identiteta Primarna operacija u diferencijalnom kalkulusu je računanje derivacije. Ova tabela sadrži derivacije nekih osnovnih funkcija . U sljedećem tekstu, f i g su diferencijabilne funkciju u skupu realnih brojeva , a c je realan broj. Ove formule su dovoljne za izračunavanje derivacija bilo koje elementarne funkcije.
Opća pravila diferenciranja Linearnost ( c f ) ′ = c f ′ {\displaystyle \left({cf}\right)'=cf'} ( f + g ) ′ = f ′ + g ′ {\displaystyle \left({f+g}\right)'=f'+g'} ( f − g ) ′ = f ′ − g ′ {\displaystyle \left({f-g}\right)'=f'-g'} Pravilo derivacije proizvoda ( f g ) ′ = f ′ g + f g ′ {\displaystyle \left({fg}\right)'=f'g+fg'} Pravilo derivacije količnika ( f g ) ′ = f ′ g − f g ′ g 2 , g ≠ 0 {\displaystyle \left({f \over g}\right)'={f'g-fg' \over g^{2},\qquad g\neq 0} Pravilo derivacije funkcije sa potencijom ( f g ) ′ = ( e g ln f ) ′ = f g ( f ′ g f + g ′ ln f ) , f > 0 {\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\qquad f>0} Pravilo derivacije složene funkcije ( f ∘ g ) ′ = ( f ′ ∘ g ) g ′ {\displaystyle (f\circ g)'=(f'\circ g)g'} Pravilo derivacije logaritma f ′ = ( ln f ) ′ f , f > 0 {\displaystyle f'=(\ln f)'f,\qquad f>0}
Derivacije jednostavnih funkcija d d x c = 0 {\displaystyle {d \over dx}c=0} d d x x = 1 {\displaystyle {d \over dx}x=1} d d x c x = c {\displaystyle {d \over dx}cx=c} d d x | x | = | x | x = sgn x , x ≠ 0 {\displaystyle {d \over dx}|x|={|x| \over x}=\operatorname {sgn} x,\qquad x\neq 0} d d x x c = c x c − 1 gdje su i x c i c x c − 1 definisane {\displaystyle {d \over dx}x^{c}=cx^{c-1}\qquad {\mbox{gdje su i }x^{c}{\mbox{ i }cx^{c-1}{\mbox{ definisane} d d x ( 1 x ) = d d x ( x − 1 ) = − x − 2 = − 1 x 2 {\displaystyle {d \over dx}\left({1 \over x}\right)={d \over dx}\left(x^{-1}\right)=-x^{-2}=-{1 \over x^{2} d d x ( 1 x c ) = d d x ( x − c ) = − c x c + 1 {\displaystyle {d \over dx}\left({1 \over x^{c}\right)={d \over dx}\left(x^{-c}\right)=-{c \over x^{c+1} d d x x = d d x x 1 2 = 1 2 x − 1 2 = 1 2 x , x > 0 {\displaystyle {d \over dx}{\sqrt {x}={d \over dx}x^{1 \over 2}={1 \over 2}x^{-{1 \over 2}={1 \over 2{\sqrt {x},\qquad x>0} d d x c x = c x ln c , c > 0 {\displaystyle {d \over dx}c^{x}={c^{x}\ln c},\qquad c>0} d d x e x = e x {\displaystyle {d \over dx}e^{x}=e^{x} d d x log c x = 1 x ln c , c > 0 , c ≠ 1 {\displaystyle {d \over dx}\log _{c}x={1 \over x\ln c},\qquad c>0,c\neq 1} d d x ln x = 1 x , x > 0 {\displaystyle {d \over dx}\ln x={1 \over x},\qquad x>0} d d x ln | x | = 1 x {\displaystyle {d \over dx}\ln |x|={1 \over x} d d x x x = x x ( 1 + ln x ) {\displaystyle {d \over dx}x^{x}=x^{x}(1+\ln x)} d d x sin x = cos x {\displaystyle {d \over dx}\sin x=\cos x} d d x cos x = − sin x {\displaystyle {d \over dx}\cos x=-\sin x} d d x tan x = sec 2 x = 1 cos 2 x {\displaystyle {d \over dx}\tan x=\sec ^{2}x={1 \over \cos ^{2}x} d d x sec x = tan x sec x {\displaystyle {d \over dx}\sec x=\tan x\sec x} d d x cot x = − csc 2 x = − 1 sin 2 x {\displaystyle {d \over dx}\cot x=-\csc ^{2}x={-1 \over \sin ^{2}x} d d x csc x = − csc x cot x {\displaystyle {d \over dx}\csc x=-\csc x\cot x} d d x arcsin x = 1 1 − x 2 {\displaystyle {d \over dx}\arcsin x={1 \over {\sqrt {1-x^{2} d d x arccos x = − 1 1 − x 2 {\displaystyle {d \over dx}\arccos x={-1 \over {\sqrt {1-x^{2} d d x arctan x = 1 1 + x 2 {\displaystyle {d \over dx}\arctan x={1 \over 1+x^{2} d d x arcsec x = 1 | x | x 2 − 1 {\displaystyle {d \over dx}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1} d d x arccot x = − 1 1 + x 2 {\displaystyle {d \over dx}\operatorname {arccot} x={-1 \over 1+x^{2} d d x arccsc x = − 1 | x | x 2 − 1 {\displaystyle {d \over dx}\operatorname {arccsc} x={-1 \over |x|{\sqrt {x^{2}-1} d d x sinh x = cosh x = e x + e − x 2 {\displaystyle {d \over dx}\sinh x=\cosh x={\frac {e^{x}+e^{-x}{2} d d x cosh x = sinh x = e x − e − x 2 {\displaystyle {d \over dx}\cosh x=\sinh x={\frac {e^{x}-e^{-x}{2} d d x tanh x = sech 2 x {\displaystyle {d \over dx}\tanh x=\operatorname {sech} ^{2}\,x} d d x sech x = − tanh x sech x {\displaystyle {d \over dx}\,\operatorname {sech} \,x=-\tanh x\,\operatorname {sech} \,x} d d x coth x = − csch 2 x {\displaystyle {d \over dx}\,\operatorname {coth} \,x=-\,\operatorname {csch} ^{2}\,x} d d x csch x = − coth x csch x {\displaystyle {d \over dx}\,\operatorname {csch} \,x=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x} d d x arcsinh x = 1 x 2 + 1 {\displaystyle {d \over dx}\,\operatorname {arcsinh} \,x={1 \over {\sqrt {x^{2}+1} d d x arccosh x = 1 x 2 − 1 {\displaystyle {d \over dx}\,\operatorname {arccosh} \,x={1 \over {\sqrt {x^{2}-1} d d x arctanh x = 1 1 − x 2 {\displaystyle {d \over dx}\,\operatorname {arctanh} \,x={1 \over 1-x^{2} d d x arcsech x = − 1 x 1 − x 2 {\displaystyle {d \over dx}\,\operatorname {arcsech} \,x={-1 \over x{\sqrt {1-x^{2} d d x arccoth x = 1 1 − x 2 {\displaystyle {d \over dx}\,\operatorname {arccoth} \,x={1 \over 1-x^{2} d d x arccsch x = − 1 | x | 1 + x 2 {\displaystyle {d \over dx}\,\operatorname {arccsch} \,x={-1 \over |x|{\sqrt {1+x^{2}
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