Question

Дифференцирование функции

Answer 1

1 Функция

$ttdisplaystyle f(x) = frac{ln(x)}{ctg(x)}\\\f'(x) = frac{(ln(x))'cdot ctg(x) - ln(x)cdot (ctg(x))'}{ctg^2(x)}=frac{displaystylefrac{1}{x}cdot ctg(x) - ln(x)cdotbigg(-frac{1}{sin^2(x)}bigg)}{ctg^2(x)}=\\\=boxed{ttfrac{displaystylefrac{ctg(x)}{x} + frac{ln(x)}{sin^2(x)}}{ctg^2(x)}}$

2 Функция

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

3 Функция

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