 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/8/a/solution

First take

F_{y} = x e^{s i n (x y)} cos (x y)

, then differentiate with respect to

x

F_{x y} = e^{s i n (x y)} [cos (x y) + x y (cos^{2} (x y) - sin (x y))] .

The function is smooth, so the two mixed partial derivatives agree.

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