Chain Rule for Partial Derivatives

• Let $u=g(x,y)$ and $v=h(x,y)$ be continuous and have first partial derivatives with respect to $x$ at a point $(x,y)$.
• Then, let $z=f(u,v)$ have continuous first partials inside a circle centered at the point $(u,v)=(g(x,y),h(x,y))$
• Then:

$$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x}$$