How was this expression differentiated?

[Auntmatilde]

This is the derivative of an expression that contains terms of the form (partial derivatives)*differentials. Are you supposed to apply the product rule to the (partial)*differential terms? And with respect to what variable?

I've tried different ways but didn't get the right result, so if anyone could tell me just the procedure I would appreciate it very much.

Thanks in advance

 

[Charles Link]

It is a little tricky writing out all of the terms of the Taylor series for a function of two variables, $f(x,y)$ in terms of $f(x_o,y_o)$, but if you proceed with the complete Taylor series to second order, you will get a term of mixed partials in the second derivative case. The $d u d v$ is $(x-x_o)(y-y_o)$ in the Taylor series...