How to differentiate the function $f(x,y)=(y,x)$ at $ a=(a_1,a_2)$?
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How to differentiate the function $f(x,y)=(y,x)$ at $ a=(a_1,a_2)$ ? $$begin{align} limlimits_{h to 0}frac{Vert f(a+h)-f(a)-Df(a)hVert}{Vert hVert} &=limlimits_{h to 0}frac{Vert (h_2+a_2,h_1+a_1) -(a_2,a_1)-(0,a_1h_1)-(a_2h_2,0)Vert}{Vert hVert}\ &=limlimits_{h to 0}frac{Vert (h_2,h_1)-(a_2h_2,a_1h_1)Vert}{Vert hVert}\ &=limlimits_{h to 0}frac{Vert (h_2(1-a_2),h_1(1-a_1))vertvert}{Vert hVert} end{align} $$ At this point I don't think I can do anything because I don't know what $a_1$ and $a_2$ are. If I knew they were between $[-1,1]$ , I could use squeeze theorem but I'm not sure what to do.
multivariable-calculus derivatives
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