Table of Derivatives Complex Functions

\(c’=0,\quad  c=const;\)

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

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

\((e^u(x))’=e^{u(x)}u’(x);\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\((\mathrm{arcth} u(x))’=\frac{1}{u^2(x)-1}u’(x),\quad x\in\mathbb{R}.\)

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