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190

capitolo viii — § 60

Il calcolo di $y'$ si esegue con le regole riassunti dai quadri seguenti:

QUADRO DELLE REGOLE DI DERIVAZIONE.

FUNZIONE	DERIVATA
$y=\varphi (x)\pm \psi (x)$	$y'=\varphi '(x)\pm \psi '(x)$
$y=\varphi (x)\psi (x)$	$y'=\varphi '(x)\psi (x)+\varphi (x)\psi '(x)$
$y={\frac {1}{\varphi (x)}}$	$y'=-{\frac {\varphi '(x)}{\left[\varphi (x)\right]^{2}}}$
$y={\frac {\psi (x)}{\varphi (x)}}$	$y={\frac {\psi '(x)\varphi (x)-\psi (x)\varphi '(x)}{\left[\varphi (x)\right]^{2}}}$
$y=f(x),x=\varphi (y)$	$f'(x)\varphi '(y)=1$
$y=f(z),z=\varphi (x)$	$y'_{x}=f'(z)\varphi (x)=y'_{z}z'_{x}$
(1) $y=\log _{e}{z}$ ; (2) $y=e^{z}$ ¹	(1) $y'={\frac {1}{z}}z'$ ; (2) $y'=e^{z}z'_{x}=yz'_{x}$
$y=\varphi (x)^{\psi (x)}$	$y'=\varphi (x)^{\psi (x)}\left[\psi '(x)\,\log \,\varphi (x)+\psi (x){\frac {\varphi (x)}{\varphi (x)}}\right]$

QUADRO DELLE DERIVATA DELLE FUNZIONI ELEMENTARI.

FUNZIONE	DERIVATA
$y={\text{costante}}$	$y'=0$
$y=x^{m}$	$y'=mx^{m-1}$
$y={\sqrt {x}}$	$y'={\frac {1}{2{\sqrt {x}}}}$
$y=\operatorname {sen} {x}$	$y'=\cos {x}$
$y=\cos {x}$	$y'=-\operatorname {sen} {x}$
$y={\text{tang}}\,x$	$y'={\frac {1}{\cos ^{2}{x}}}$
$y=\cot {x}$	$y'=-{\frac {1}{\operatorname {sen} ^{2}{x}}}$
$y=\log {x}$	$y'={\frac {1}{x}}$
$y=\log _{a}{x}$	$y'={\frac {1}{x}}\log _{a}{e}$
$y=a^{x}$	$y'=a^{x}\log _{e}{a}$
$y=e^{x}$	$y'=e^{x}$
$y={\text{arc}}{\begin{matrix}\operatorname {sen} \\\cos \end{matrix}}x$	$y'=\pm {\frac {1}{\sqrt {1-x^{2}}}}$
$y={\text{arc tang}}\,x$	$y'={\frac {1}{1+x^{2}}}$