Question

Найти производные функции

Answer 1

$1)\; \; y=arctg\sqrt{x^2-1}\\\\y'=\frac{1}{1+(x^2-1)}\cdot \frac{1}{2\sqrt{x^2-1}}\cdot 2x\\\\2)\; \; y=e^{3x}-2x\, tg3x\\\\y'=3e^{3x}-(2\, tg3x+2x\cdot \frac{3}{cos^23x})\\\\3)\; \; y=x^{tg4x}\\\\lny=tg4x\cdot lnx\\\\\frac{y'}{y}=\frac{4}{cos^24x}\cdot lnx+tg4x\cdot \frac{1}{x}\\\\y'=x^{tg4x}\cdot \Big (\frac{4\, lnx}{cos^24x}+\frac{tg4x}{x}\Big )$

$4)\; \; x^4+xy^2-7y=8\\\\4x^3+(y^2+2xyy')-7y'=0\\\\y'\cdot (2xy-7)=-4x^3-y^2\\\\y'=-\frac{4x^3+y^2}{2xy-7}$