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Solution

f(x)=sin(xy)+xcos(y)
f_x= (sin(xy))\'+(xcos(y))\' where derivative is with respect to x

=cos(xy)*(xy)\'+cos(y) chain rule for first term

=cos(xy)*y+cos(y)

f_y=(sin(xy))\'+(xcos(y))\' where derivative is with respect to y

=cos(xy)*(xy)\'-xsin(y)

=cos(xy)*x-xsin(y)

f_xy=(f_x)_y or (f_y)_x (they are equal)

=(cos(xy)y)\'+(cos(y))\' where derivative is with respect to y

=(cos(xy))\'y+cos(xy)y\'-sin(y) product rule

=-sin(xy)*(xy)\'y+cos(xy)-sin(y)   chain rule

=-sin(xy)*xy+cos(xy)-sin(y)