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