Дифференцирование функции

1 Функция

$\tt\displaystyle f(x) = \frac{ln(x)}{ctg(x)}\\\\\\f'(x) = \frac{(ln(x))'\cdot ctg(x) - ln(x)\cdot (ctg(x))'}{ctg^2(x)}=\frac{\displaystyle\frac{1}{x}\cdot ctg(x) - ln(x)\cdot\bigg(-\frac{1}{sin^2(x)}\bigg)}{ctg^2(x)}=\\\\\\=\boxed{\tt\frac{\displaystyle\frac{ctg(x)}{x} + \frac{ln(x)}{sin^2(x)}}{ctg^2(x)}}$

2 Функция

$\tt\displaystyle f(x) = e^{15x - 8}\\\\\\f'(x) = e^{15x - 8}\cdot (15x - 8)' = \boxed{\tt 15e^{15x - 8}}$

3 Функция

$\tt\displaystyle f(x) = \frac{4}{x^8}-\sqrt[14]{\tt x^5}-4 = 4\cdot x^{-8} - x^{\displaystyle\frac{5}{14}}-4\\\\\\f'(x) = -8\cdot 4\cdot x^{-9} - \frac{5}{14}\cdot x^{\displaystyle\frac{5}{14} - 1}=-\frac{32}{x^9}-\frac{5\cdot x^{\displaystyle -\frac{9}{14}}}{14}=\boxed{\tt\frac{32}{x^9}-\frac{5}{14\cdot \sqrt[14]{\tt x^9}}}$