Let $f,h,l: \mathbb{R}^2 \to \mathbb{R}$ be derivable functions. If $g(x,y)=f(h(x,y),l(x,y))$, is the following formula true? $$\frac{∂g}{∂x}(x,y)= \left(\frac{∂f}{∂x}(h(x,y),l(x,y)) +\frac{∂f}{∂y}(h(x,y),l(x,y))\right)\left( \frac{∂h}{∂x}(x,y)+\frac{∂l}{∂x}(x,y)\right) $$ It looks close to the chain rule $\frac{∂f}{∂x} \cdot \frac{∂x}{∂t}$
Also does this notation $\frac{∂f}{∂x}(h(x,y),l(x,y)) $ make sense as a derivative of $f(x,y)$ by the variable $x$ and after the derivation, substituting $h,l$ in the place of $x,y$ ?
If $g(x,y)=f(h(x,y),l(x,y)) $ The actual answer, as suggested by @Git Gud is
$$\frac{∂g}{∂x}(x,y)= \left(\frac{∂f}{∂x}(h(x,y),l(x,y))\frac{∂h}{∂x}(x,y) +\frac{∂f}{∂y}(h(x,y),l(x,y))\frac{∂l}{∂x}(x,y)\right) $$
More info on the link below:
http://en.wikipedia.org/wiki/Chain_rule#Higher_dimensions