Table of Derivatives Formulas

\(c’=0,\quad  c=const;\)

\((x^\alpha)’=\alpha x^{\alpha-1}, \quad x\in \mathbb{R}, \alpha\in
\mathbb{R};\)

\((a^x)’=a^x\ln a,\quad a>0, a\neq 1, x\in \mathbb{R};\)

\((e^x)’=e^x;\)

\((\log_a x)’=\frac{1}{x\ln a}, \quad x>0;\)

\((\log_a|x|)’=\frac{1}{x\ln a},\quad x\neq 0;\)

\((\ln x)’=\frac{1}{x},\quad x>0;\)

\((\sin x)’=\cos x, \quad x\in \mathbb{R};\)

\((\cos x)’=-\sin x\quad x\in \mathbb{R};\)

\((\mathrm{tg} x)’=\frac{1}{\cos^2 x},\quad x\neq \frac{\pi}{2}(2n+1), n\in
\mathbb{Z};\)

\((\mathrm{ctg} x)’=-\frac{1}{\sin^2 x},\quad x\neq \pi n, n\in
\mathbb{Z};\)

\((\arcsin x)’=\frac{1}{\sqrt{1-x^2}},\quad |x|<1;\)

\((\arccos  x)’=-\frac{1}{\sqrt{1-x^2}},\quad |x|<1;\)

\((\mathrm{arctg} x)’=\frac{1}{1+x^2},\quad x\in\mathbb{R};\)

\((\mathrm{arcctg} x)’=-\frac{1}{1+x^2},\quad x\in\mathbb{R};\)

\((\mathrm{sh} x)’=\mathrm{ch} x, \quad x\in \mathbb{R};\)

\((\mathrm{ch} x)’=\mathrm{sh} x\quad x\in \mathbb{R};\)

\((\mathrm{th} x)’=\frac{1}{\mathrm{ch}^2 x},\quad x\neq \frac{\pi}{2}(2n+1), n\in
\mathbb{Z};\)

\((\mathrm{cth} x)’=-\frac{1}{\mathrm{sh}^2 x},\quad x\neq \pi n, n\in
\mathbb{Z};\)

\((\mathrm{arsh} x)’=\frac{1}{\sqrt{x^2+1}},\quad x\in\mathbb{R};\)

\((\mathrm{arch}  x)’=-\frac{1}{\sqrt{x^2-1}},\quad |x|>1;\)

\((\mathrm{arth} x)’=\frac{1}{1-x^2},\quad x\in\mathbb{R};\)

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