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

$\displaystyle \frac{\partial f}{\partial x}i=(\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x})i$

Is my logic correct?