11 14.2236fx is the partial derivativeWrt x and we treatysty as a constant vice versafx fy is partial wrt xfx x fx y fy x fylyf x y Yy 6 93 2 34 Compute fxx fxy fyxfyyfxx fx Zyx 643 2 2y fxy Fy Zyx 643 2fyx fx x2 18xy 3 2 1842 2 1892fyy fy x2 1842 336 9Iff is a function oftwo variables andthe partialsare continuous thenfxy fyxCorollary All partial derivatives of a certain order n areequaHowever must be the same number ofx's andy's i e fxxy fyxx I f xxx 3615f.ttTtsinwycwcoscautlsinwxUxsin out sin wx wsinant coscoxUxx w sin out cos wx w's inwxsin antpUtt ow cos cwt sin w x ow sin w x sin at2b Ux Ex ex's sin x y m sin x y text cos xyUxx Fy ext sin xy tros xy ex sin xy tcos x yayy Yy ext sin xy toosexy cos x y sinceyII31 All same 32 Two same one diff 3 3 fxxyfyxz.IO II3 All different I
