Таблица производных

Формулы^[1][2].

${\displaystyle f(x)}$	${\displaystyle f'(x)}$	Условия
${\displaystyle c}$	${\displaystyle 0}$	Любое ${\displaystyle x}$
${\displaystyle x^{a}}$	${\displaystyle ax^{a-1}}$	${\displaystyle x>0}$ для произвольного вещественного ${\displaystyle a}$
${\displaystyle {\sqrt {x}}}$	${\displaystyle 1/(2{\sqrt {x}})}$	${\displaystyle x>0}$
${\displaystyle 1/x}$	${\displaystyle -1/x^{2}}$	${\displaystyle x\neq 0}$
${\displaystyle |x|}$	${\displaystyle x/|x|}$	${\displaystyle x\neq 0}$
${\displaystyle e^{x}}$	${\displaystyle e^{x}}$	Любое ${\displaystyle x}$
${\displaystyle a^{x}}$	${\displaystyle a^{x}\ln a}$	${\displaystyle a>0}$
${\displaystyle \ln x}$	${\displaystyle 1/x}$	${\displaystyle x>0}$
${\displaystyle \ln |x|}$	${\displaystyle 1/x}$	${\displaystyle x\neq 0}$
${\displaystyle \log _{a}x}$	${\displaystyle 1/(x\ln a)}$	${\displaystyle x>0,\ a>0,\ a\neq 1}$

Формулы^[3][4].

${\displaystyle f(x)}$	${\displaystyle f'(x)}$	Условия
${\displaystyle \sin x}$	${\displaystyle \cos x}$	Любое ${\displaystyle x}$
${\displaystyle \cos x}$	${\displaystyle -\sin x}$	Любое ${\displaystyle x}$
${\displaystyle \operatorname {tg} x}$	${\displaystyle 1/\cos ^{2}x}$	${\displaystyle \cos x\neq 0}$
${\displaystyle \operatorname {ctg} x}$	${\displaystyle -1/\sin ^{2}x}$	${\displaystyle \sin x\neq 0}$
${\displaystyle \arcsin x}$	${\displaystyle 1/{\sqrt {1-x^{2}}}}$	${\displaystyle |x|<1}$
${\displaystyle \arccos x}$	${\displaystyle -1/{\sqrt {1-x^{2}}}}$	${\displaystyle |x|<1}$
${\displaystyle \operatorname {arctg} x}$	${\displaystyle 1/(1+x^{2})}$	Любое ${\displaystyle x}$
${\displaystyle \operatorname {arcctg} x}$	${\displaystyle -1/(1+x^{2})}$	Любое ${\displaystyle x}$

Формулы^[5].

${\displaystyle f(x)}$	${\displaystyle f'(x)}$	Условия
${\displaystyle \operatorname {sh} x}$	${\displaystyle \operatorname {ch} x}$	Любое ${\displaystyle x}$
${\displaystyle \operatorname {ch} x}$	${\displaystyle \operatorname {sh} x}$	Любое ${\displaystyle x}$
${\displaystyle \operatorname {th} x}$	${\displaystyle 1/\operatorname {ch} ^{2}x}$	Любое ${\displaystyle x}$
${\displaystyle \operatorname {cth} x}$	${\displaystyle -1/\operatorname {sh} ^{2}x}$	${\displaystyle x\neq 0}$

Для положительного целого ${\displaystyle a}$ формула производной ${\displaystyle x^{a}}$ действует также при отрицательных ${\displaystyle x}$. В точке ${\displaystyle x=0}$ производная равна единице при ${\displaystyle a=1}$ и нулю при ${\displaystyle a>1}$. При ${\displaystyle a=0}$ производная постоянной функции равна нулю^[1]. Для ${\displaystyle \operatorname {arcctg} x}$ выбрана ветвь со значениями в ${\displaystyle (0,\pi )}$.

Правила применяют в точках, где функции определены и дифференцируемы^[1][2].

Формула	Правило или условие
${\displaystyle (cu)'=cu'}$	Постоянный множитель
${\displaystyle (u\pm v)'=u'\pm v'}$	Сумма и разность
${\displaystyle (uv)'=u'v+uv'}$	Произведение
${\displaystyle \left({\frac {u}{v}}\right)'={\frac {u'v-uv'}{v^{2}}}}$	${\displaystyle v\neq 0}$
${\displaystyle (f(u))'=f'(u)u'}$	Сложная функция
${\displaystyle (\ln |u|)'=u'/u}$	${\displaystyle u\neq 0}$

Здесь ${\displaystyle n}$ — положительное целое число, ${\displaystyle a}$ и ${\displaystyle b}$ — постоянные^[6].

${\displaystyle f(x)}$	${\displaystyle f^{(n)}(x)}$	Условия
${\displaystyle x^{a}}$	${\displaystyle a(a-1)\cdots (a-n+1)x^{a-n}}$	${\displaystyle x>0}$
${\displaystyle e^{ax+b}}$	${\displaystyle a^{n}e^{ax+b}}$	Любое ${\displaystyle x}$
${\displaystyle \sin(ax+b)}$	${\displaystyle a^{n}\sin(ax+b+\pi n/2)}$	Любое ${\displaystyle x}$
${\displaystyle \cos(ax+b)}$	${\displaystyle a^{n}\cos(ax+b+\pi n/2)}$	Любое ${\displaystyle x}$
${\displaystyle \ln |x|}$	${\displaystyle (-1)^{n-1}(n-1)!/x^{n}}$	${\displaystyle x\neq 0}$