(1)
z=(x+y)sin(\(z^{2}\)+2xy+\(y^{2}\)) は
z=(x+y)sin(\(x^{2}\)+2xy+\(y^{2}\)) の誤りと思われます。

z=(x+y)sin(x+y\()^{2}\)
∂z/∂x=sin(x+y\()^{2}\)+(x+y)2(x+y)cos(x+y\()^{2}\)
=sin(x+y\()^{2}\)+2(x+y\()^{2}\)cos(x+y\()^{2}\)
x+y，(x+y\()^{2}\)は対象式だから
∂z/∂y=∂z/∂x=sin(x+y\()^{2}\)+2(x+y\()^{2}\)cos(x+y\()^{2}\)

(2)
u=log(\(x^{2}\)+\(y^{2}\)+\(z^{2}\)+2xy+2yz+2zx)
=2log(x+y+z)
∂u/∂x=∂u/∂y=∂u/∂z=2/(x+y+z)