Правила дифференцирования функций

$1^o. \quad \left({cf}\right)' = cf'$

$2^o. \quad \left({f + g}\right)' = f' + g'$

$3^o. \quad \left({f - g}\right)' = f' - g'$

$4^o. \quad \left({fg}\right)' = f'g + fg'$

$5^o. \quad \left({f \over g}\right)' = {f'g - fg' \over g^2}, \quad g \ne 0$

$6^o. \quad (f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\quad f > 0$

$7^o. \quad (f (g(x)))' = f'(g(x))\cdot g'(x)$

$8^o. \quad f' = (\ln f)'f, \quad f > 0$

$9^o. \quad (f^c)' = c\left(f^{c-1}\right)f'$

Производные элементарных функций

$1^o. \quad (c)' = 0$

$2^o. \quad \left(x\right)' = 1$

$3^o. \quad \left(cx\right)' = c$

$4^o. \quad \left(x^c\right)' = cx^{c-1}$, ҳангоми $x^c$ и $cx^{c-1}$ муайян будан ва $c \ne 0$

$5^o. \quad \left(|x|\right)' = {x \over |x|} = \operatorname{sgn}\,x,\quad x \ne 0$

$6^o. \quad \left({1 \over x}\right)' = \left(x^{-1}\right)' = -x^{-2} = -{1 \over x^2}$

$7^o. \quad \left({1 \over x^c}\right)' = \left(x^{-c}\right)' = -{c \over x^{c+1}}$

$8^o. \quad \left(\sqrt{x}\right)' = \left(x^{1\over 2}\right)' = {1 \over 2} x^{-{1\over 2}}  = {1 \over 2 \sqrt{x}}, \quad x > 0$

$9^o. \quad \left(\sqrt [n] {x}\right)' = \left(x^{1\over n}\right)' = {1 \over n} x^{1-n\over n} = \frac {1} {n \cdot \sqrt [n] {x^{n-1}}}$

$10^o. \quad \left(c^x\right)' = c^x \ln c,\quad c > 0$

$11^o. \quad \left(e^x\right)' = e^x$

$12^o. \quad \left(e^{f(x)}\right)' = f'(x)e^{f(x)}$

$13^o. \quad \left(\ln x\right)' = {1 \over x}$

$14^o. \quad \left(\log_a x\right)' = \frac{\log_a e} {x} =  \frac {1} {x \ln a }$

$15^o. \quad \left(\log_a f(x)\right)' = \left(\frac {\ln f(x)}{\ln(a)}\right)' = \frac{ f'(x) }{ f(x) \ln(a)}$

$16^o. \quad \left(\sin x\right)' = \cos x$

$17^o. \quad \left(\cos x\right)'  = -\sin x$

$18^o. \quad \left(\operatorname{tg}\,x\right)' = \sec^2 x = { 1 \over \cos^2 x} = \operatorname{tg}^2 x + 1$

$19^o. \quad \left(\operatorname{ctg}\,x\right)' = -\,\operatorname{cosec}^2\,x = { -1 \over \sin^2 x}$

$20^o. \quad \left(\sec x\right)' =\,\operatorname{tg}x \,\sec x$

$21^o. \quad \left(\,\operatorname{cosec}\,x\right)' = -\,\operatorname{ctg}x \,\operatorname{cosec}\,x$

$22^o. \quad \left(\arcsin x\right)' = { 1 \over \sqrt{1 - x^2}}$

$23^o. \quad \left(\arccos x\right)' = {-1 \over \sqrt{1 - x^2}}$

$24^o. \quad \left(\,\operatorname{arctg}\,x\right)' = { 1 \over 1 + x^2}$

$25^o. \quad \left(\,\operatorname{arcctg}\,x\right)' = {-1 \over 1 + x^2}$

$26^o. \quad \left(\,\operatorname{arcsec}\,x\right)' = { 1 \over |x|\sqrt{x^2 - 1}}$

$27^o. \quad \left(\,\operatorname{arccosec}\,x\right)' = {-1 \over |x|\sqrt{x^2 - 1}}$