ПРОИЗВОДНЫЕ

Таблица производных основных элементарных функций:

$${C}'=0, C=const$$ $$ {(x^\alpha)}' = \alpha x^{\alpha -1}$$

$$ {(a^x)}'= a^x \cdot \ln(a)$$

$$ {(e^x)}'=e^x$$

$$ {(\log_a (x))}'= \frac1 {x \cdot \ln(a)}$$

$$ {(\ln(x))}'= \frac1 {x}$$

$$ {(\sin(x))}'= \cos (x)$$

$$ {(\cos(x))}'=-\sin (x)$$

$$ {(\sqrt x)}'= \frac1 {2 \cdot \sqrt x}$$

$$ {(\textrm{tg} (x))}'= \frac1 {\cos^2(x)}$$

$$ {(\textrm{ctg} (x))}'= - \frac1 {\sin^2(x)}$$

$$ {(\textrm{arcsin} (x))}'= \frac1 {\sqrt {1-x^2}}$$

$$ {(\textrm{arccos} (x))}'= - \frac1 {\sqrt {1-x^2}}$$

$$ {(\textrm{arctg} (x))}'= \frac1 {1+x^2}$$

$$ {(\textrm{arcctg} (x))}'= - \frac1 {1+x^2}$$

$$ {(\textrm{sh} (x))}'= \textrm{ch}(x)$$

$$ {(\textrm{ch} (x))}'= \textrm{sh}(x)$$

$$ {(\textrm{th} (x))}'= \frac1 {\textrm{ch}^2(x)}$$

$$ {(\textrm{cth} (x))}'= \frac1 {\textrm{sh}^2(x)}$$

Правила дифференцирования (производная произведения, частного и композиции функций)

$$ {(u \cdot v)}'= {u}' \cdot v + u \cdot {v}', \; {\left ( \frac{u}{v} \right )}'=\frac {{u}' \cdot v - u \cdot {v}'}{v^2}$$

$$ {(f(g(x)))}'={f}'(y) | _{y=g(x)} \cdot {g}'(x)$$