Why create a profile on Shaalaacom?U}{\partial x \partial y} \; There's a factor of 2 missing in all your second derivatives The result is exactly as you'd expect The variable you're differentiating with respect to, matters If it's x, then y is treated as a constant, and vice versa So if the "active" variable is leading in the numerator in one derivative, the same should apply in the other
How To Do Implicit Differentiation 7 Steps With Pictures