Multivariable chain rule derivative definition calculation

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How can we calculate such expressions, in particular:

Let $f,g$ be both diffrentiable, how to calculate this limit:

$$\lim_{h \to 0} \frac{f(x(t) +hg(x(t)), y(t)+h\frac{dg}{dx} \frac{dx}{dt}) -f(x(t), y(t) )}h\ $$