Taula de derivadesarticle de llista de Wikimedia / From Wikipedia, the free encyclopedia En el procés de càlcul de derivades o diferenciació, es pot obtenir la derivada de qualsevol funció elemental emprant les regles de derivació i la taula de derivades de les funcions base a partir de les quals es construeixen la resta de funcions elementals. Les derivades d'aquestes funcions base s'obtenen normalment a partir de la definició de derivada, aplicant les propietats de cada funció i amb les tècniques de càlcul de límits. Taula de derivades Més informació , ... Funció F: primitiva de f Funció f: derivada de F Funcions elementals f ( x ) = k {\displaystyle f(x)=k\,} f ′ ( x ) = 0 {\displaystyle f'(x)=0\,} f ( x ) = x {\displaystyle f(x)=x\,} f ′ ( x ) = 1 {\displaystyle f'(x)=1\,} f ( x ) = x n {\displaystyle f(x)=x^{n}\,} f ′ ( x ) = n x n − 1 {\displaystyle f'(x)=nx^{n-1}\,} f ( x ) = x {\displaystyle f(x)={\sqrt {x}}\,} f ′ ( x ) = 1 2 x {\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}\,} f ( x ) = e x {\displaystyle f(x)=e^{x}\,} f ′ ( x ) = e x {\displaystyle f'(x)=e^{x}\,} f ( x ) = ln ( x ) {\displaystyle f(x)=\ln(x)\,} f ′ ( x ) = 1 x {\displaystyle f'(x)={\frac {1}{x}}\,} f ( x ) = a x (amb a > 0 ) {\displaystyle f(x)=a^{x}\quad {\text{(amb }}a>0)\,} f ′ ( x ) = a x ln ( a ) {\displaystyle f'(x)=a^{x}\ln(a)\,} f ( x ) = log b ( x ) {\displaystyle f(x)=\log _{b}(x)\,} f ′ ( x ) = 1 x ln ( b ) {\displaystyle f'(x)={\frac {1}{x\ln(b)}}\,} f ( x ) = 1 x n = ( x n ) − 1 = x − n {\displaystyle f(x)={\frac {1}{x^{n}}}=(x^{n})^{-1}=x^{-n}\,} f ′ ( x ) = − n x − n − 1 {\displaystyle f'(x)=-nx^{-n-1}\,} Funcions trigonomètriques f ( x ) = sin ( x ) {\displaystyle f(x)=\sin(x)\,} f ′ ( x ) = cos ( x ) {\displaystyle f'(x)=\cos(x)\,} f ( x ) = cos ( x ) {\displaystyle f(x)=\cos(x)\,} f ′ ( x ) = − sin ( x ) {\displaystyle f'(x)=-\sin(x)\,} f ( x ) = tg ( x ) {\displaystyle f(x)=\operatorname {tg} (x)\,} f ′ ( x ) = sec 2 ( x ) {\displaystyle f'(x)=\sec ^{2}(x)\,} f ( x ) = sec ( x ) {\displaystyle f(x)=\sec(x)\,} f ′ ( x ) = sec ( x ) tg ( x ) {\displaystyle f'(x)=\sec(x)\operatorname {tg} (x)\,} f ( x ) = cosec ( x ) {\displaystyle f(x)=\operatorname {cosec} (x)\,} f ′ ( x ) = − cosec ( x ) cotg ( x ) {\displaystyle f'(x)=-\operatorname {cosec} (x)\operatorname {cotg} (x)\,} f ( x ) = cotg ( x ) {\displaystyle f(x)=\operatorname {cotg} (x)\,} f ′ ( x ) = − cosec 2 ( x ) {\displaystyle f'(x)=-\operatorname {cosec} ^{2}(x)\,} f ( x ) = arcsin ( x ) {\displaystyle f(x)=\arcsin(x)\,} f ′ ( x ) = 1 1 − x 2 {\displaystyle f'(x)={\frac {1}{\sqrt {1-x^{2}}}}\,} f ( x ) = arccos ( x ) {\displaystyle f(x)=\arccos(x)\,} f ′ ( x ) = − 1 1 − x 2 {\displaystyle f'(x)={\frac {-1}{\sqrt {1-x^{2}}}}\,} f ( x ) = arctg ( x ) {\displaystyle f(x)=\operatorname {arctg} (x)\,} f ′ ( x ) = 1 1 + x 2 {\displaystyle f'(x)={\frac {1}{1+x^{2}}}\,} Funcions hiperbòliques f ( x ) = sinh x {\displaystyle f(x)=\sinh x\,} f ′ ( x ) = cosh x = e x + e − x 2 {\displaystyle f'(x)=\cosh x={\frac {e^{x}+e^{-x}}{2}}\,} f ( x ) = arsinh x {\displaystyle f(x)=\operatorname {arsinh} \,x\,} f ′ ( x ) = 1 x 2 + 1 {\displaystyle f'(x)={1 \over {\sqrt {x^{2}+1}}}\,} f ( x ) = cosh x {\displaystyle f(x)=\cosh x\,} f ′ ( x ) = sinh x = e x − e − x 2 {\displaystyle f'(x)=\sinh x={\frac {e^{x}-e^{-x}}{2}}\,} f ( x ) = arcosh x {\displaystyle f(x)=\operatorname {arcosh} \,x\,} f ′ ( x ) = 1 x 2 − 1 {\displaystyle f'(x)={\frac {1}{\sqrt {x^{2}-1}}}\,} f ( x ) = tgh x {\displaystyle f(x)=\operatorname {tgh} \,x\,} f ′ ( x ) = sech 2 x {\displaystyle f'(x)={\operatorname {sech} ^{2}\,x}\,} f ( x ) = artgh x {\displaystyle f(x)=\operatorname {artgh} \,x\,} f ′ ( x ) = 1 1 − x 2 {\displaystyle f'(x)={1 \over 1-x^{2}}\,} f ( x ) = sech x {\displaystyle f(x)=\operatorname {sech} \,x\,} f ′ ( x ) = − tgh x sech x {\displaystyle f'(x)=-\operatorname {tgh} \,x\,\operatorname {sech} \,x\,} f ( x ) = arsech x {\displaystyle f(x)=\operatorname {arsech} \,x\,} f ′ ( x ) = − 1 x 1 − x 2 {\displaystyle f'(x)=-{1 \over x{\sqrt {1-x^{2}}}}\,} f ( x ) = cosech x {\displaystyle f(x)=\operatorname {cosech} \,x\,} f ′ ( x ) = − cotgh x cosech x {\displaystyle f'(x)=-\,\operatorname {cotgh} \,x\,\operatorname {cosech} \,x\,} f ( x ) = arcosech x {\displaystyle f(x)=\operatorname {arcosech} \,x} f ′ ( x ) = − 1 | x | 1 + x 2 {\displaystyle f'(x)=-{1 \over |x|{\sqrt {1+x^{2}}}}\,} f ( x ) = cotgh x {\displaystyle f(x)=\operatorname {cotgh} \,x\,} f ′ ( x ) = − cosech 2 x {\displaystyle f'(x)=-\,\operatorname {cosech} ^{2}\,x\,} f ( x ) = arcotgh x {\displaystyle f(x)=\operatorname {arcotgh} \,x\,} f ′ ( x ) = 1 1 − x 2 {\displaystyle f'(x)={1 \over 1-x^{2}}\,} Tanca Funcions especials Funció Gamma ( Γ ( x ) ) ′ = ∫ 0 ∞ t x − 1 e − t ln t d t {\displaystyle (\Gamma (x))'=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt} ( Γ ( x ) ) ′ = Γ ( x ) ( ∑ n = 1 ∞ ( ln ( 1 + 1 n ) − 1 x + n ) − 1 x ) = Γ ( x ) ψ ( x ) {\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)} Funció zeta de Riemann ( ζ ( x ) ) ′ = − ∑ n = 1 ∞ ln n n x = − ln 2 2 x − ln 3 3 x − ln 4 4 x − ⋯ {\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 \!} ( ζ ( x ) ) ′ = − ∑ p primer p − x ln p ( 1 − p − x ) 2 ∏ q primer , q ≠ p 1 1 − q − x {\displaystyle (\zeta (x))'=-\sum _{p{\text{ primer}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ primer}},q\neq p}{\frac {1}{1-q^{-x}}}\!}
En el procés de càlcul de derivades o diferenciació, es pot obtenir la derivada de qualsevol funció elemental emprant les regles de derivació i la taula de derivades de les funcions base a partir de les quals es construeixen la resta de funcions elementals. Les derivades d'aquestes funcions base s'obtenen normalment a partir de la definició de derivada, aplicant les propietats de cada funció i amb les tècniques de càlcul de límits. Taula de derivades Més informació , ... Funció F: primitiva de f Funció f: derivada de F Funcions elementals f ( x ) = k {\displaystyle f(x)=k\,} f ′ ( x ) = 0 {\displaystyle f'(x)=0\,} f ( x ) = x {\displaystyle f(x)=x\,} f ′ ( x ) = 1 {\displaystyle f'(x)=1\,} f ( x ) = x n {\displaystyle f(x)=x^{n}\,} f ′ ( x ) = n x n − 1 {\displaystyle f'(x)=nx^{n-1}\,} f ( x ) = x {\displaystyle f(x)={\sqrt {x}}\,} f ′ ( x ) = 1 2 x {\displaystyle f'(x)={\frac {1}{2{\sqrt {x}}}}\,} f ( x ) = e x {\displaystyle f(x)=e^{x}\,} f ′ ( x ) = e x {\displaystyle f'(x)=e^{x}\,} f ( x ) = ln ( x ) {\displaystyle f(x)=\ln(x)\,} f ′ ( x ) = 1 x {\displaystyle f'(x)={\frac {1}{x}}\,} f ( x ) = a x (amb a > 0 ) {\displaystyle f(x)=a^{x}\quad {\text{(amb }}a>0)\,} f ′ ( x ) = a x ln ( a ) {\displaystyle f'(x)=a^{x}\ln(a)\,} f ( x ) = log b ( x ) {\displaystyle f(x)=\log _{b}(x)\,} f ′ ( x ) = 1 x ln ( b ) {\displaystyle f'(x)={\frac {1}{x\ln(b)}}\,} f ( x ) = 1 x n = ( x n ) − 1 = x − n {\displaystyle f(x)={\frac {1}{x^{n}}}=(x^{n})^{-1}=x^{-n}\,} f ′ ( x ) = − n x − n − 1 {\displaystyle f'(x)=-nx^{-n-1}\,} Funcions trigonomètriques f ( x ) = sin ( x ) {\displaystyle f(x)=\sin(x)\,} f ′ ( x ) = cos ( x ) {\displaystyle f'(x)=\cos(x)\,} f ( x ) = cos ( x ) {\displaystyle f(x)=\cos(x)\,} f ′ ( x ) = − sin ( x ) {\displaystyle f'(x)=-\sin(x)\,} f ( x ) = tg ( x ) {\displaystyle f(x)=\operatorname {tg} (x)\,} f ′ ( x ) = sec 2 ( x ) {\displaystyle f'(x)=\sec ^{2}(x)\,} f ( x ) = sec ( x ) {\displaystyle f(x)=\sec(x)\,} f ′ ( x ) = sec ( x ) tg ( x ) {\displaystyle f'(x)=\sec(x)\operatorname {tg} (x)\,} f ( x ) = cosec ( x ) {\displaystyle f(x)=\operatorname {cosec} (x)\,} f ′ ( x ) = − cosec ( x ) cotg ( x ) {\displaystyle f'(x)=-\operatorname {cosec} (x)\operatorname {cotg} (x)\,} f ( x ) = cotg ( x ) {\displaystyle f(x)=\operatorname {cotg} (x)\,} f ′ ( x ) = − cosec 2 ( x ) {\displaystyle f'(x)=-\operatorname {cosec} ^{2}(x)\,} f ( x ) = arcsin ( x ) {\displaystyle f(x)=\arcsin(x)\,} f ′ ( x ) = 1 1 − x 2 {\displaystyle f'(x)={\frac {1}{\sqrt {1-x^{2}}}}\,} f ( x ) = arccos ( x ) {\displaystyle f(x)=\arccos(x)\,} f ′ ( x ) = − 1 1 − x 2 {\displaystyle f'(x)={\frac {-1}{\sqrt {1-x^{2}}}}\,} f ( x ) = arctg ( x ) {\displaystyle f(x)=\operatorname {arctg} (x)\,} f ′ ( x ) = 1 1 + x 2 {\displaystyle f'(x)={\frac {1}{1+x^{2}}}\,} Funcions hiperbòliques f ( x ) = sinh x {\displaystyle f(x)=\sinh x\,} f ′ ( x ) = cosh x = e x + e − x 2 {\displaystyle f'(x)=\cosh x={\frac {e^{x}+e^{-x}}{2}}\,} f ( x ) = arsinh x {\displaystyle f(x)=\operatorname {arsinh} \,x\,} f ′ ( x ) = 1 x 2 + 1 {\displaystyle f'(x)={1 \over {\sqrt {x^{2}+1}}}\,} f ( x ) = cosh x {\displaystyle f(x)=\cosh x\,} f ′ ( x ) = sinh x = e x − e − x 2 {\displaystyle f'(x)=\sinh x={\frac {e^{x}-e^{-x}}{2}}\,} f ( x ) = arcosh x {\displaystyle f(x)=\operatorname {arcosh} \,x\,} f ′ ( x ) = 1 x 2 − 1 {\displaystyle f'(x)={\frac {1}{\sqrt {x^{2}-1}}}\,} f ( x ) = tgh x {\displaystyle f(x)=\operatorname {tgh} \,x\,} f ′ ( x ) = sech 2 x {\displaystyle f'(x)={\operatorname {sech} ^{2}\,x}\,} f ( x ) = artgh x {\displaystyle f(x)=\operatorname {artgh} \,x\,} f ′ ( x ) = 1 1 − x 2 {\displaystyle f'(x)={1 \over 1-x^{2}}\,} f ( x ) = sech x {\displaystyle f(x)=\operatorname {sech} \,x\,} f ′ ( x ) = − tgh x sech x {\displaystyle f'(x)=-\operatorname {tgh} \,x\,\operatorname {sech} \,x\,} f ( x ) = arsech x {\displaystyle f(x)=\operatorname {arsech} \,x\,} f ′ ( x ) = − 1 x 1 − x 2 {\displaystyle f'(x)=-{1 \over x{\sqrt {1-x^{2}}}}\,} f ( x ) = cosech x {\displaystyle f(x)=\operatorname {cosech} \,x\,} f ′ ( x ) = − cotgh x cosech x {\displaystyle f'(x)=-\,\operatorname {cotgh} \,x\,\operatorname {cosech} \,x\,} f ( x ) = arcosech x {\displaystyle f(x)=\operatorname {arcosech} \,x} f ′ ( x ) = − 1 | x | 1 + x 2 {\displaystyle f'(x)=-{1 \over |x|{\sqrt {1+x^{2}}}}\,} f ( x ) = cotgh x {\displaystyle f(x)=\operatorname {cotgh} \,x\,} f ′ ( x ) = − cosech 2 x {\displaystyle f'(x)=-\,\operatorname {cosech} ^{2}\,x\,} f ( x ) = arcotgh x {\displaystyle f(x)=\operatorname {arcotgh} \,x\,} f ′ ( x ) = 1 1 − x 2 {\displaystyle f'(x)={1 \over 1-x^{2}}\,} Tanca Funcions especials Funció Gamma ( Γ ( x ) ) ′ = ∫ 0 ∞ t x − 1 e − t ln t d t {\displaystyle (\Gamma (x))'=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt} ( Γ ( x ) ) ′ = Γ ( x ) ( ∑ n = 1 ∞ ( ln ( 1 + 1 n ) − 1 x + n ) − 1 x ) = Γ ( x ) ψ ( x ) {\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)} Funció zeta de Riemann ( ζ ( x ) ) ′ = − ∑ n = 1 ∞ ln n n x = − ln 2 2 x − ln 3 3 x − ln 4 4 x − ⋯ {\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 \!} ( ζ ( x ) ) ′ = − ∑ p primer p − x ln p ( 1 − p − x ) 2 ∏ q primer , q ≠ p 1 1 − q − x {\displaystyle (\zeta (x))'=-\sum _{p{\text{ primer}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ primer}},q\neq p}{\frac {1}{1-q^{-x}}}\!}