Differentials and implicit differentiation Consider this example. Suppose $x$ is a function of two variables $s$ and $t$,
$$x = \sin(s+t)$$
Taking the differential as in doing implicit differentiation [1],
$$dx = \cos(s+t)(ds+dt) = \cos(s+t)dt + \cos(s+t)ds$$
I know the right way of taking the differential of $x$ is by
$dx = \dfrac{\partial x}{\partial s}ds + \dfrac{\partial x}{\partial t}dt$ [2]
But why do the above method [1], which I cannot make any sense of mathematically, give the same result as [2]?
I do not understand method [1] because $s$ and $t$ are supposed to be independent variables not functions to be differentiated. i.e. $ds$ just means $s-s_0$, same with $dt$.

This is a special case: you may consider a simple substitution $y = s + t$
$x = \sin(y)$
$dx = \cos(y)dy$
$dx = \cos(s+y)(ds + dt)$
if you do this with $x=s^t$
$y = s^t$
$x = y$
$dx = dy$
$dx = ts^{t-1}ds + s^t\ln(s)dt$
which isn't useful at all...
