Question

Найдите производную
(tgx)^(6x)

Answer 1

$y=(tgx)^{6x}\\\\lny=6x\cdot ln(tgx)\\\\\frac{y'}{y}=6\cdot ln(tgx)+6x\cdot \frac{1}{tgx}\cdot \frac{1}{cos^2x}=6\cdot ln(tgx)+\frac{6x}{sinx\cdot cosx}=\\\\ =6\cdot ln(tgx)+\frac{12x}{sin2x}\\\\y'=y\cdot \Big (6\cdot ln(tgx)+\frac{6x}{sinx\cdot cosx}\Big )\\\\y'=(tgx)^{6x}\cdot \Big (6\cdot ln(tgx)+\frac{6x}{sinx\cdot cosx}\Big )$