If
$f=f(g(x),h(x))$
Then I can easily demonstrate the chain rule:
$\frac{df}{dx}=\frac{\partial{f}}{\partial{g}}\cdot\frac{\partial{g}}{\partial{x}}+\frac{\partial{f}}{\partial{h}}\cdot\frac{\partial{h}}{\partial{x}}$
But what if
$f=f(x,g(x,t))$
Then it'll be wrong if I write the chain rule this way:
$\frac{\partial{f}}{\partial{x}}=\frac{\partial{f}}{\partial{x}}+\frac{\partial{f}}{\partial{g}}\cdot\frac{\partial{g}}{\partial{x}}$
because the $\frac{\partial{f}}{\partial{x}}$ in the left has a different meaning from the one in the right.
Then how do I express this equation?