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 \(c \ne 0\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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