Formulario sulle derivate

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Regole di derivazione

Siano f(x) e g(x) funzioni reali di variabile reale x derivabili, e sia \mathrm{D} l'operazione di derivazione rispetto a x:

\mathrm{D}[f(x)]=f'(x) \qquad \mathrm{D}[g(x)]=g'(x)

\displaystyle D[af(x) + bg(x) ]=a\cdot f'(x) + b\cdot g'(x) Derivata della somma
\displaystyle \mathrm{D}[{f(x) \cdot g(x)}] = f'(x) \cdot g(x) + f(x) \cdot g'(x) Derivata del prodotto
\mathrm{D}\! \displaystyle \left[ {f(x) \over g(x)} \right] = { f'(x) \cdot g(x) - f(x) \cdot g'(x) \over (g(x))^2} Derivata del quoziente
\mathrm{D}\!\displaystyle \left[ {1 \over f(x)} \right] = -{f'(x) \over (f(x))^2} Derivata della funzione reciproca
\mathrm{D}\displaystyle \left[ f \left(g(x) \right)\right] = f' \left(g(x) \right) \cdot g'(x) Derivata delle funzioni composte
\mathrm{D}\displaystyle\left[ f(x)^{g(x)} \right] = f(x)^{g(x)}\left[ g'(x)\ln(f(x)) + \frac{g(x)\cdot f'(x)}{f(x)} \right] Derivata della potenza

Derivate di funzioni elementari

Funzione Derivata
\displaystyle y = k \displaystyle y' = 0
\displaystyle y = x^n \displaystyle y'= n\,x^{(n-1)}
\displaystyle y = x \displaystyle y' = 1
\displaystyle y = \frac{1}{x}=x^{-1} \displaystyle y' = -\frac{1}{x^2}
\displaystyle y = \sqrt{x}=x^\frac{1}{2} \displaystyle y' = \frac{1}{2\sqrt{x}}
\displaystyle y = \sqrt[n]{x^m}=x^ \frac{m}{n} \displaystyle y'= \frac{m}{n}\cdot x^{\frac{m}{n}-1}
\displaystyle y=|x| \displaystyle y' = \frac{|x|}{x}
\displaystyle y=\log_a{x} \displaystyle y'=\frac{1}{x}\log_a{e}=\frac{1}{x}\frac{1}{\ln{a}}
\displaystyle y=\ln{x} \displaystyle y'=\frac{1}{x}
\displaystyle y=a^{x} \displaystyle y'=a^{x}\ln{a}
\displaystyle y=e^{x} \displaystyle y'=e^{x}
\displaystyle y=\sin{x} \displaystyle y'=\cos{x}
\displaystyle y=\cos{x} \displaystyle y'=-\sin{x}
\displaystyle y=\tan{x} \displaystyle y'=\frac{1}{\cos^2{x}}=1+\tan^2{x}
\displaystyle y=\cot{x} \displaystyle y'=-\frac{1}{\sin^2{x}}=-(1+cotan^2{x})
\displaystyle y=\arcsin{x} \displaystyle y'=\frac{1}{\sqrt{1-x^2}}
\displaystyle y=\arccos{x} \displaystyle y'=-\frac{1}{\sqrt{1-x^2}}
\displaystyle y=\arctan{x} \displaystyle y'=\frac{1}{1+x^2}
\displaystyle y=arccot x \displaystyle y'=-\frac{1}{1+x^2}
\displaystyle y=\sinh x \displaystyle y'=\cosh x
\displaystyle y=\cosh x \displaystyle y'=\sinh x
\displaystyle y=\tanh x \displaystyle y'=1 - \tanh^2 x

 Derivate di funzioni composte

Funzione Derivata
\displaystyle y =f(x)^n \displaystyle y'= n\,f(x)^{(n-1)} \cdot f'(x)
\displaystyle y=|f(x)| \displaystyle y' = \frac{|f(x)|}{f(x)}\cdot f'(x)
\displaystyle y=\ln{|f(x)|} \displaystyle y'=\frac{1}{f(x)}\cdot f'(x)
\displaystyle y=a^{f(x)} \displaystyle y'=a^{f(x)}\ln{a}\cdot f'(x)
\displaystyle y=e^{f(x)} \displaystyle y'=e^{f(x)}\cdot f'(x)
\displaystyle y=\sin{f(x)} \displaystyle y'=\cos{f(x)}\cdot f'(x)
\displaystyle y=\cos{f(x)} \displaystyle y'=-\sin{f(x)}\cdot f'(x)
\displaystyle y=\tan{f(x)} \displaystyle y'=\frac{1}{\cos^2{f(x)}}\cdot f'(x)
\displaystyle y=\cot{f(x)} \displaystyle y'=-\frac{1}{\sin^2{f(x)}}\cdot f'(x)
\displaystyle y=\arcsin{f(x)} \displaystyle y'=\frac{1}{\sqrt{1-f(x)^2}}\cdot f'(x)
\displaystyle y=\arccos{f(x)} \displaystyle y'=-\frac{1}{\sqrt{1-f(x)^2}}
\displaystyle y=\arctan{f(x)}  \displaystyle y'=\frac{1}{1+f(x)^2}\cdot f'(x)
\displaystyle y=arccot f(x)  \displaystyle y'=-\frac{1}{1+f(x)^2}\cdot f'(x)

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