I need to show that $\nabla(au+bv)=a\nabla u + b\nabla v$ where $a,b$ are constants and $u=u(x,y),v=(x,y)$ are differentiale real-valued functions. If I have a function $f(u,v)=au+bv$ where $u,v$ are functions, then ultimately $f(u,v)=f(x,y)$ right? Then I could right the $i$ component of the gradient:
$\frac{\partial f}{\partial x}i=(\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x})i$