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

Question

zale101 · Answer

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

zale101 · Answer

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

amistre64 · Answer

are we to see if Fxy = Fyx ?

amistre64 · Answer

2nd order partial derivatives are of the form

Fxx
Fxy
Fyx
Fyy

the descriptor 'mixed' refers to xy or yx mixture

zale101 · Answer

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

zale101 · Answer

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