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
| \(y=f(x)\) | \(y=f'(x)\) |
| \(C\) | \(0\) |
| \(x\) | \(1\) |
| \(x^n\) | \(nx^{n-1}\) |
| \(\sqrt{x}\) | \(\displaystyle\frac{1}{2\sqrt{x}}\) |
| \(\displaystyle\frac{1}{x}\) | \(-\displaystyle\frac{1}{x^2}\) |
| \(\sin x\) | \(\cos x\) |
| \(\cos x\) | \(-\sin x\) |
| \(\mathrm{tg} x\) | \(\displaystyle\frac{1}{\cos^2 x}\) |
| \(\mathrm{ctg} x\) | \(-\displaystyle\frac{1}{\sin^2 x} \) |
| \(a^x\) | \(a^x\ln a\) |
| \(\ln x\) | \(\displaystyle\frac{1}{x}\) |
| \(\log_ax\) | \(\displaystyle\frac{1}{x\ln a}\) |
| \(e^x\) | \(e^x\) |
| \(\arcsin x\) | \(\displaystyle\frac{1}{\sqrt{1-x^2}}\) |
| \(\arccos x\) | \(-\displaystyle\frac{1}{\sqrt{1-x^2}}\) |
| \(\mathrm{arctg} x\) | \(\displaystyle\frac{1}{1+x^2}\) |
| \(\mathrm{arcctg} x\) | \(-\displaystyle\frac{1}{1+x^2}\) |
| \(\mathrm{sh} x\) | \(\mathrm{ch} x\) |
| \(\mathrm{ch} x\) | \(\mathrm{sh} x\) |
| \(\mathrm{th} x\) | \(\displaystyle\frac{1}{\mathrm{ch}^2 x}\) |
| \(\mathrm{cth} x\) | \(-\displaystyle\frac{1}{\mathrm{sh}^2 x}\) |
| \(\mathrm{sec} x\) | \(\mathrm{tg}{x}\cdot\mathrm{sec}x\) |
| \(\mathrm{cosec} x\) | \(-\mathrm{ctg}{x}\cdot\mathrm{cosec}x\) |
Производные высших порядков
\((x^m)^{(n)}=m(m-1)\cdot…\cdot(m-n+1)x^{m-n}\)
\((a^x)^{(n)}=a^x\mathrm{ln}^na\), \(a>0\)
\((e^x)^{(n)}=e^x\)
\((\sin{x})^{(n)}=\sin(x+\displaystyle\frac{n\pi}{2})\)
\((\cos{x})^{(n)}=\cos(x+\displaystyle\frac{n\pi}{2})\)
\((\mathrm{ln}{x})^{(n)}=\displaystyle\frac{(-1)^{n-1}(n-1)!}{x^n}\)
Формула Лейбница
\((f\cdot{g})^{(n)}=C_n^{0}f^{(0)}g^{(n)}+C_n^1f^{(1)}g^{(n-1)}+…+C_n^nf^{(n)}g^{(0)}\)