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Производная функции

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Правила дифференцирования

\((C\cdot f(x))’=C\cdot f'(x)\)

\((f(x)\pm g(x))’=f'(x)\pm g'(x)\)

\((f(x)\cdot g(x))’=f'(x)\cdot g(x)+f(x)\cdot g'(x)\)

\(\left(\displaystyle\frac{f(x)}{g(x)}\right)’=\displaystyle\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{g^2(x)}\)

\((g(f(x)))’=g'(f(x))\cdot f'(x)\)

Производные элементарных функций

\(y=f(x)\) \(y=f'(x)\)
\(C\) \(0\)
\(x\) \(1\)
\(x^n\) \(nx^{n-1}\)
\(\sqrt{x}\) \(\displaystyle\frac{1}{2\sqrt{x}}\)
\(\displaystyle\frac{1}{x}\) \(-\displaystyle\frac{1}{x^2}\)
\(\sin x\) \(\cos x\)
\(\cos x\) \(-\sin x\)
\(\mathrm{tg} x\) \(\displaystyle\frac{1}{\cos^2 x}\)
\(\mathrm{ctg} x\) \(-\displaystyle\frac{1}{\sin^2 x} \)
\(a^x\) \(a^x\ln a\)
\(\ln x\) \(\displaystyle\frac{1}{x}\)
\(\log_ax\) \(\displaystyle\frac{1}{x\ln a}\)
 \(e^x\) \(e^x\)
 \(\arcsin x\)  \(\displaystyle\frac{1}{\sqrt{1-x^2}}\)
 \(\arccos x\) \(-\displaystyle\frac{1}{\sqrt{1-x^2}}\)
 \(\mathrm{arctg} x\) \(\displaystyle\frac{1}{1+x^2}\)
\(\mathrm{arcctg} x\) \(-\displaystyle\frac{1}{1+x^2}\)
 \(\mathrm{sh} x\) \(\mathrm{ch} x\)
 \(\mathrm{ch} x\) \(\mathrm{sh} x\)
 \(\mathrm{th} x\) \(\displaystyle\frac{1}{\mathrm{ch}^2 x}\)
 \(\mathrm{cth} x\) \(-\displaystyle\frac{1}{\mathrm{sh}^2 x}\)
 \(\mathrm{sec} x\) \(\mathrm{tg}{x}\cdot\mathrm{sec}x\)
 \(\mathrm{cosec} x\) \(-\mathrm{ctg}{x}\cdot\mathrm{cosec}x\)