Question

Найти производные dy/dx заданных функций,k=1

Answer 1

$1)\; \; y=ln\sqrt{1+e^{2x}}\\\\y'=\frac{1}{\sqrt{1+e^{2x}}}\cdot \frac{1}{2\sqrt{1+e^{2x}}}\cdot 2e^{2x}=\frac{e^{2x}}{1+e^{2x}}$

$2)\; \; \left \{ {{x=cos^32t} \atop {y=sin^32t}} \right. \; \; \; y'_{x}=\frac{y'_{t}}{x'_{t}}\\\\y'_{t}=3\, sin^22t\cdot cos2t\cdot 2\\\\x'_{t}=3\, cos^22t\cdot (-sin2t)\cdot 2\\\\y'_{x}=\frac{3\, sin^22t\cdot cos2t\cdot 2}{-3\, cos^22t\cdot sin2t\cdot 2}=-\frac{sin2t}{cos2t}=-tg2t$