Question

find indicated partial derivate
f(x, y) = xy(x 2 − y 2 ) / x 2 + y 2 (x,y) ≠(0,0)

=0 (x,y)≠(0,0)

; fx(0, 0) and fy(0, 0)​

Answer 1

Answer:

Using the chain rule with u = xy for the partial derivatives of cos(xy) ∂ ∂x cos(xy) = ∂ cos(u) ∂u ∂u ∂x = − sin(u)y = −y sin(xy) , ∂ ∂y cos(xy) = ∂ cos(u) ∂u ∂u ∂y = − sin(u)x = −x sin(xy) . Thus the partial derivatives of z = sin(x) cos(xy) are ∂z ∂x = cos(xy) cos(x) − y sin(x) sin(xy) , ∂z ∂y = −x sin(x) sin(xy) .