Because of the implicit function theorem, is it in general true that:
$$ \frac{\partial f}{\partial x} = \frac{\partial f}{\partial a} \frac{\partial a}{\partial x} + \frac{\partial f}{\partial b} \frac{\partial b}{\partial x} + \cdots + \frac{\partial f}{\partial n} \frac{\partial n}{\partial x}$$
Where $f = f(a,b,c, \dots n)$ and $a=a(x),\ b=b(x)\ , \dots ,\ n=n(x)$ ?