Question

14. Consider the function f(x, y) = yx^4 * e^x². Find fxyxyx (x, y).​

Answer 1

The second partial derivatives of
`f` exist everywhere in its domain. By Schwarz's theorem, the mixed second-order partials derivatives are equal:

`f_(xy) = f_(yx)`

Then

`f(x,y) = y x^4 e^(x^2) \\\\ \implies f_y = x^4 e^(x^2) \\\\ \implies f_(yx) = f_(xy) = g(x) \\\\ \implies f_(xyx) = g'(x) \\\\ \implies f_(xyxy) = 0 \\\\ \implies f_(xyxyx) = \boxed{0}`

We don't actually need to compute
`g` and
`g'` to know that they are both free of
`y`. So when we differentiate for a second time with respect to
`y`, the whole thing cancels out.