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**Opalg** 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 $\displaystyle y^2x^2$, then you use the product rule to get $\displaystyle \frac d{dx}(y^2x^2) = 2y\frac{dy}{dx}*x^2 + y^2*2x.$ 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 $\displaystyle y^2$ with respect to x, you have to use the chain rule, differentiating it first with respect to y and then multiplying by $\displaystyle \frac{dy}{dx}.$

The calculation for $\displaystyle \frac{\partial}{\partial x}\bigl(\frac{\partial w}{\partial s}\frac{\partial s}{\partial x}\bigr)$ is just the analogous procedure for functions of two variables.