Question

Найти модуль градиента функции z=sin(y/x) в точке (1;0)

Answer 1

$z = sin\frac{y}{x}\\z'_x = cos(\frac{y}{x}) * (\frac{y}{x})'_x = - \frac{y* cos(\frac{y}{x})}{x^2}\\z'_y = cos(\frac{y}{x}) * (\frac{y}{x})'_y = \frac{1}{x}*cos(\frac{y}{x})\\grad(z) = (- \frac{y* cos(\frac{y}{x})}{x^2}; \frac{1}{x}*cos(\frac{y}{x}))\\ |grad(z)| = \sqrt{\frac{y^2*cos^2(\frac{y}{x})}{x^4} + \frac{cos^2(\frac{y}{x}) }{x^2}} = \sqrt{\frac{cos^2(\frac{y}{x})(x^2+y^2)}{x^4}} = \frac{|cos(\frac{y}{x})|}{x^2}\sqrt{x^2+y^2}$

$x = 1\\y = 0\\|grad(z)| = \frac{|cos(\frac{0}{1})|}{1^2}\sqrt{1^2+0^2} = cos0 = 1\\Answer: 1$