Question

Помогите решить задание
найти полный дифференциал функции z(x,y)=ctg^2(xy)+2^(y/x)

Answer 1

$z(x,y)=ctg^2(xy) + 2^\frac{y}{x}\\z'_x = (ctg^2(xy) + 2^\frac{y}{x})'_x = 2ctg(xy)*(-\frac{1}{sin^2(xy)})*(xy)'_x + 2^\frac{y}{x}*(\frac{y}{x})'_x = \frac{-2y*ctg(xy)}{sin^2(xy)} - \frac{y*2^\frac{y}{x}}{x^2}\\z'_y = (ctg^2(xy) + 2^\frac{y}{x})'_y = 2ctg(xy)*(-\frac{1}{sin^2(xy)})*(xy)'_y + 2^\frac{y}{x}*(\frac{y}{x})'_y = \frac{-2x*ctg(xy)}{sin^2(xy)} + \frac{2^\frac{y}{x}}{x}$

$dz = (\frac{-2y*ctg(xy)}{sin^2(xy)} - \frac{y*2^\frac{y}{x}}{x^2})dx + (\frac{-2x*ctg(xy)}{sin^2(xy)} + \frac{2^\frac{y}{x}}{x})dy$