Question

Confirm that the mixed second-order partial derivatives of f are the same.

Answer 1

\(\large f(x,y)=4x^2-8xy^4+7y^5-3 \)

Answer 2

\(f_x=8x-8y^4\)
\(f_{y}=-32xy^3+35y^4\)

Answer 3

are we to see if Fxy = Fyx ?

Answer 4

2nd order partial derivatives are of the form

Fxx
Fxy
Fyx
Fyy

the descriptor 'mixed' refers to xy or yx mixture

Answer 5

I forgot to close the question. I found my way through this question.
Thanks!

Answer 6

\(\Large \frac{ ∂^2f }{ ∂ x∂ y }=\frac{ ∂ }{ ∂x }(\frac{ ∂f }{ ∂ y })=\frac{ ∂ }{∂ x }(−32xy^3+35y^4)=−32y^3\)
\(\Large \frac{ ∂^2f }{ ∂ y∂ x }=\frac{ ∂ }{ ∂y }(\frac{ ∂f }{ ∂ x })=\frac{ ∂ }{∂ y}(−8x−8y^4)=−32y^3\)