In matematica , le regole di derivazione e le derivate fondamentali sono regole studiate per evitare di dover calcolare ogni volta il limite del rapporto incrementale di funzioni , e utilizzate al fine di facilitare la derivazione di funzioni di maggiore complessità.
Siano f ( x ) {\displaystyle f(x)} e g ( x ) {\displaystyle g(x)} funzioni reali di variabile reale x {\displaystyle x} derivabili, e sia D {\displaystyle \mathrm {D} } l'operazione di derivazione rispetto a x {\displaystyle x} :
D [ f ( x ) ] = f ′ ( x ) , D [ g ( x ) ] = g ′ ( x ) . {\displaystyle \mathrm {D} [f(x)]=f'(x),\qquad \mathrm {D} [g(x)]=g'(x).} D [ α f ( x ) + β g ( x ) ] = α f ′ ( x ) + β g ′ ( x ) , α , β ∈ R . {\displaystyle \mathrm {D} [\alpha f(x)+\beta g(x)]=\alpha f'(x)+\beta g'(x),\qquad \alpha ,\beta \in \mathbb {R} .} D [ f ( x ) ⋅ g ( x ) ] = f ′ ( x ) ⋅ g ( x ) + f ( x ) ⋅ g ′ ( x ) . {\displaystyle \mathrm {D} [{f(x)\cdot g(x)}]=f'(x)\cdot g(x)+f(x)\cdot g'(x).} D [ f ( x ) g ( x ) ] = f ′ ( x ) ⋅ g ( x ) − f ( x ) ⋅ g ′ ( x ) g ( x ) 2 . {\displaystyle \mathrm {D} \left[{f(x) \over g(x)}\right]={f'(x)\cdot g(x)-f(x)\cdot g'(x) \over g(x)^{2}}.} D [ 1 f ( x ) ] = − f ′ ( x ) f ( x ) 2 . {\displaystyle \mathrm {D} \left[{1 \over f(x)}\right]=-{f'(x) \over f(x)^{2}}.} D [ f − 1 ( x ) ] = 1 f ′ ( f − 1 ( x ) ) . {\displaystyle \mathrm {D} [f^{-1}(x)]={1 \over f'(f^{-1}(x))}.} D [ f ( g ( x ) ) ] = f ′ ( g ( x ) ) ⋅ g ′ ( x ) . {\displaystyle \mathrm {D} \left[f\left(g(x)\right)\right]=f'\left(g(x)\right)\cdot g'(x).} D [ f ( x ) g ( x ) ] = f ( x ) g ( x ) [ g ′ ( x ) ln ( f ( x ) ) + g ( x ) f ′ ( x ) f ( x ) ] . {\displaystyle \mathrm {D} \left[f(x)^{g(x)}\right]=f(x)^{g(x)}\left[g'(x)\ln(f(x))+{\frac {g(x)f'(x)}{f(x)}}\right].} Ognuna di queste funzioni, se non altrimenti specificato, è derivabile in tutto il suo campo di esistenza .
D ( a ) = 0 , a costante . {\displaystyle \mathrm {D} (a)=0,\qquad a{\text{ costante}}.} D ( x ) = 1. {\displaystyle \mathrm {D} (x)=1.} D ( a x ) = a , a costante . {\displaystyle \mathrm {D} (ax)=a,\qquad a{\text{ costante}}.} D ( x 2 ) = 2 x . {\displaystyle \mathrm {D} (x^{2})=2x.} D ( x 3 ) = 3 x 2 . {\displaystyle \mathrm {D} (x^{3})=3x^{2}.} Più in generale si ha:
D ( x n ) = n x n − 1 , con n ∈ N . {\displaystyle \mathrm {D} (x^{n})=nx^{n-1},\qquad {\text{con }}n\in \mathbb {N} .} Da quest'ultima relazione segue che se f ( x ) {\displaystyle f(x)} è un polinomio generico di grado n {\displaystyle n} , allora D ( f ( x ) ) {\displaystyle D\left(f(x)\right)} è in generale un polinomio di grado n − 1 {\displaystyle n-1} .
D ( x α ) = α x α − 1 , con α ∈ R . {\displaystyle \mathrm {D} (x^{\alpha })=\alpha x^{\alpha -1},\qquad {\text{con }}\alpha \in \mathbb {R} .} D ( x 2 ) = 1 2 x 2 . {\displaystyle \mathrm {D} ({\sqrt[{2}]{x}})={\frac {1}{2{\sqrt[{2}]{x}}}}.} D ( x m n ) = m n x m − n n , se x > 0. {\displaystyle \mathrm {D} ({\sqrt[{n}]{x^{m}}})={{\frac {m}{n}}{\sqrt[{n}]{x^{m-n}}}},\qquad {\text{se }}x>0.} D ( | x | ) = | x | x = x | x | . {\displaystyle \mathrm {D} (|x|)={\dfrac {|x|}{x}}={\dfrac {x}{|x|}}.} D ( log b x ) = log b e x = 1 x ln b . {\displaystyle \mathrm {D} (\log _{b}x)={\frac {\log _{b}\mathrm {e} }{x}}={\frac {1}{x\ln b}}.} D ( ln x ) = 1 x . {\displaystyle \mathrm {D} (\ln x)={\frac {1}{x}}.} D ( e x ) = e x . {\displaystyle \mathrm {D} (e^{x})=\mathrm {e} ^{x}.} D ( a x ) = a x ln a . {\displaystyle \mathrm {D} (a^{x})=a^{x}\ln a.} D ( x x ) = x x ( 1 + ln x ) . {\displaystyle \mathrm {D} (x^{x})=x^{x}(1+\ln x).} D ( sin x ) = cos x . {\displaystyle \mathrm {D} (\sin x)=\cos x.} D ( cos x ) = − sin x . {\displaystyle \mathrm {D} (\cos x)=-\sin x.} D ( tan x ) = 1 + tan 2 x = 1 cos 2 x . {\displaystyle \mathrm {D} (\tan x)=1+\tan ^{2}x={1 \over \cos ^{2}x}.} D ( cot x ) = − ( 1 + cot 2 x ) = − 1 sin 2 x . {\displaystyle \mathrm {D} (\cot x)=-(1+\cot ^{2}x)=-{\frac {1}{\sin ^{2}x}}.} D ( sec x ) = tan x sec x . {\displaystyle \mathrm {D} (\sec x)=\tan x\sec x.} D ( csc x ) = − cot x csc x . {\displaystyle \mathrm {D} (\csc x)=-\cot x\csc x.} D ( arcsin x ) = 1 1 − x 2 . {\displaystyle \mathrm {D} (\arcsin x)={\frac {1}{\sqrt {1-x^{2}}}}.} D ( arccos x ) = − 1 1 − x 2 . {\displaystyle \mathrm {D} (\arccos x)=-{\frac {1}{\sqrt {1-x^{2}}}}.} D ( arctan x ) = 1 1 + x 2 . {\displaystyle \mathrm {D} (\arctan x)={\frac {1}{1+x^{2}}}.} D ( arccot x ) = − 1 1 + x 2 . {\displaystyle \mathrm {D} (\operatorname {arccot} x)={-1 \over 1+x^{2}}.} D ( arcsec x ) = 1 | x | x 2 − 1 . {\displaystyle \mathrm {D} (\operatorname {arcsec} x)={1 \over |x|{\sqrt {x^{2}-1}}}.} D ( arccsc x ) = − 1 | x | x 2 − 1 . {\displaystyle \mathrm {D} (\operatorname {arccsc} x)={-1 \over |x|{\sqrt {x^{2}-1}}}.} D ( sinh x ) = cosh x . {\displaystyle \mathrm {D} (\sinh x)=\cosh x.} D ( cosh x ) = sinh x . {\displaystyle \mathrm {D} (\cosh x)=\sinh x.} D ( tanh x ) = 1 − tanh 2 x = 1 cosh 2 x . {\displaystyle \mathrm {D} (\tanh x)=1-\tanh ^{2}x={1 \over \cosh ^{2}x}.} D ( coth x ) = − csch 2 x . {\displaystyle \mathrm {D} ({\mbox{coth}}\,x)=-{\mbox{csch}}^{2}\,x.} D ( sech x ) = − tanh x sech x . {\displaystyle \mathrm {D} ({\mbox{sech}}\,x)=-\tanh x\;{\mbox{sech}}\,x.} D ( csch x ) = − coth x csch x . {\displaystyle \mathrm {D} ({\mbox{csch}}\,x)=-{\mbox{coth}}\,x\;{\mbox{csch}}\,x.} D ( settsinh x ) = 1 x 2 + 1 . {\displaystyle \mathrm {D} ({\mbox{settsinh}}\,x)={1 \over {\sqrt {x^{2}+1}}}.} D ( settanh x ) = 1 1 − x 2 . {\displaystyle \mathrm {D} ({\mbox{settanh}}\,x)={1 \over 1-x^{2}}.} D ( settcoth x ) = 1 1 − x 2 . {\displaystyle \mathrm {D} ({\mbox{settcoth}}\,x)={1 \over 1-x^{2}}.} D ( settsech x ) = − 1 x 1 − x 2 . {\displaystyle \mathrm {D} ({\mbox{settsech}}\,x)={-1 \over x{\sqrt {1-x^{2}}}}.} D ( settcsch x ) = − 1 | x | 1 + x 2 . {\displaystyle \mathrm {D} ({\mbox{settcsch}}\,x)={-1 \over |x|{\sqrt {1+x^{2}}}}.} D ( | f ( x ) | ) = f ′ ( x ) f ( x ) | f ( x ) | = f ′ ( x ) | f ( x ) | f ( x ) . {\displaystyle \mathrm {D} (|f(x)|)=f'(x){\dfrac {f(x)}{|f(x)|}}=f'(x){\dfrac {|f(x)|}{f(x)}}.} D ( [ f ( x ) ] n ) = n ⋅ f ( x ) n − 1 ⋅ f ′ ( x ) . {\displaystyle \mathrm {D} ([f(x)]^{n})=n\cdot f(x)^{n-1}\cdot f'(x).} D ( ln f ( x ) ) = f ′ ( x ) f ( x ) . {\displaystyle \mathrm {D} (\ln f(x))={f'(x) \over f(x)}.} D ( ln | f ( x ) | ) = f ′ ( x ) f ( x ) . {\displaystyle \mathrm {D} (\ln |f(x)|)={f'(x) \over f(x)}.} D ( e f ( x ) ) = e f ( x ) ⋅ f ′ ( x ) . {\displaystyle \mathrm {D} (\mathrm {e} ^{f(x)})=\mathrm {e} ^{f(x)}\cdot f'(x).} D ( a f ( x ) ) = a f ( x ) ⋅ f ′ ( x ) ⋅ ln a . {\displaystyle \mathrm {D} (a^{f(x)})=a^{f(x)}\cdot f'(x)\cdot \ln a.} D ( sin f ( x ) ) = cos f ( x ) ⋅ f ′ ( x ) . {\displaystyle \mathrm {D} (\sin f(x))=\cos f(x)\cdot f'(x).} D ( cos f ( x ) ) = − sin f ( x ) ⋅ f ′ ( x ) . {\displaystyle \mathrm {D} (\cos f(x))=-\sin f(x)\cdot f'(x).} D ( tan f ( x ) ) = f ′ ( x ) cos 2 f ( x ) . {\displaystyle \mathrm {D} (\tan f(x))={f'(x) \over \cos ^{2}f(x)}.} D ( arcsin f ( x ) ) = f ′ ( x ) 1 − [ f ( x ) ] 2 . {\displaystyle D(\arcsin f(x))={f'(x) \over {\sqrt {1-[f(x)]^{2}}}}.} D ( arccos f ( x ) ) = − f ′ ( x ) 1 − [ f ( x ) ] 2 . {\displaystyle D(\arccos f(x))={-f'(x) \over {\sqrt {1-[f(x)]^{2}}}}.} D ( arctan f ( x ) ) = f ′ ( x ) 1 + [ f ( x ) ] 2 . {\displaystyle D(\arctan f(x))={f'(x) \over 1+[f(x)]^{2}}.} D ( f ( x ) g ( x ) ) = f ( x ) g ( x ) ⋅ [ g ′ ( x ) ⋅ ln f ( x ) + g ( x ) ⋅ f ′ ( x ) f ( x ) ] . {\displaystyle D(f(x)^{g(x)})=f(x)^{g(x)}\cdot \left[g'(x)\cdot \ln f(x)+g(x)\cdot {f'(x) \over f(x)}\right].} Dimostrazione
f ( x ) g ( x ) = e ln f ( x ) g ( x ) = e g ( x ) ⋅ ln f ( x ) {\displaystyle {f(x)^{g(x)}}={e^{{\ln }{f(x)^{g(x)}}}}={e^{g(x)\cdot {{\ln }f(x)}}}} e dunque si deriva seguendo la regola di D ( e f ( x ) ) {\displaystyle D({e^{f(x)}})} e del prodotto. D ( x f ( x ) ) = x f ( x ) ⋅ [ f ′ ( x ) ⋅ ln x + f ( x ) x ] . {\displaystyle D(x^{f(x)})=x^{f(x)}\cdot \left[f'(x)\cdot \ln x+{f(x) \over x}\right].}