That is correct, as far as it goes. In fact, w is not "directly" a function of x. It is a function of s (and t), which are in turn functions of x and y. So it is indirectly a function of x (and y), and to differentiate it partially with respect to x you have to use the chain rule. The other term in the product, s(x), is directly a function of x, so you can differentiate that with respect to x without any further work.

You should already be familiar with procedures of this sort when dealing with functions of a single variable, if you have ever used the technique of implicit differentiation. For example, if y is a function of x and you want to differentiate an expression like

, then you use the product rule to get

In that calculation, you have a product of two functions, a function of y and a function of x. You can differentiate it, using the product rule. But in order to differentiate

with respect to x, you have to use the chain rule, differentiating it first with respect to y and then multiplying by

The calculation for

is just the analogous procedure for functions of two variables.