Given some differentiable function $f(u(x,y),v(x,y))$ I am trying to prove that $\displaystyle \frac{\partial f}{\partial x} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x}$. My question is whether my proof (below) is correct.
Attempted proof: By definition $\displaystyle \frac{\partial f}{\partial x} = \lim_{x\to a}\frac{f(u(x,y),v(x,y))-f(u(a,y),v(a,y))}{x-a} = \lim_{x\to a}\frac{f(u(x,y),v(x,y))-f(u(a,y),v(x,y))+f(u(a,y),v(x,y))-f(u(a,y),v(a,y))}{x-a} = \lim_{x\to a}\frac{f(u(x,y),v(x,y))-f(u(a,y),v(x,y))}{u(x,y)-u(a,y)} \dot{}\frac{u(x,y)-u(a,y)}{x-a} + \lim_{x\to a}\frac{f(u(a,y),v(x,y))-f(u(a,y),v(a,y))}{v(x,y)-v(a,y)} \dot{}\frac{v(x,y)-v(a,y)}{x-a} = \lim_{x\to a}\frac{f(u(x,y),v(x,y))-f(u(a,y),v(x,y))}{u(x,y)-u(a,y)} \dot{}\lim_{x\to a}\frac{u(x,y)-u(a,y)}{x-a} + \lim_{x\to a}\frac{f(u(a,y),v(x,y))-f(u(a,y),v(a,y))}{v(x,y)-v(a,y)} \dot{}\lim_{x\to a}\frac{v(x,y)-v(a,y)}{x-a} = \frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x}$