Question

найти производную сложной функции

Answer 1

$z=arcctg\frac{x}{y}\; \; ,\; \; \; x=e^{2t}+1\; \; ,\; \; y=e^{2t}-1\; ;\\\\\\\frac{dz}{dt}=\frac{\partial z}{\partial x}\cdot \frac{dx}{dt}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dt}\\\\\\\frac{dz}{dt}=-\frac{1}{1+\frac{x^2}{y^2}}\cdot \frac{1}{y}\cdot2e^{2t}-\frac{1}{1+\frac{x^2}{y^2}}\cdot (-\frac{x}{y^2})\cdot 2e^{2t}=-\frac{y^2}{x^2+y^2}\cdot  \frac{2e^{2t}}{y}+\frac{y^2}{x^2+y^2}\cdot \frac{2x\, e^{2t}}{y^2}=\\\\=2e^{2t}\cdot (-\frac{y}{x^2+y^2}+\frac{x}{x^2+y^2y})=\frac{2e^{2t}}{x^2+y^2}\cdot (x-y)$