The Taylor expansion for a function of two variables (up to the second order) is
#f(x,y) = f(x_0,y_0)#
#qquad +((del f)/(del x))_{(x_0,y_0)}(x-x_0)+((del f)/(del y))_{(x_0,y_0)}(y-y_0)#
#+1/(2!)((del^2 f)/(del x^2))_{(x_0,y_0)}(x-x_0)^2+1/(2!)((del^2 f)/(del y^2))_{(x_0,y_0)}(y-y_0)^2#
#qquad +((del^2 f)/(del x del y))_{(x_0,y_0)}(x-x_0)(y-y_0)+...#
For the function
#f(x) = xy^2+cos(xy)#
the relevant derivatives are
#(del f)/(del x) = y^2-ysin(xy),qquad ((del f)/(del x))_{(1,pi/2)} = pi^2/4-pi/2 #
#(del f)/(del y) = 2xy-xsin(xy),qquad ((del f)/(del y))_{(1,pi/2)} = pi-1 #
#(del^2 f)/(del x^2) = -y^2cos(xy),qquad ((del^2 f)/(del x^2))_{(1,pi/2)} = 0 #
#(del^2 f)/(del y^2) =2x -x cos(xy),qquad ((del^2 f)/(del y^2))_{(1,pi/2)} = 2 #
#(del^2 f)/(del x del y) = 2y-xycos(xy)-sin
(xy),qquad ((del^2 f)/(del xdely))_{(1,pi/2)} = pi-1 #
Combining all these (and the result that #f(1,pi/2) = pi^2/4#) we get (up to second order)
#f(x,y) = pi^2/4 + (pi^2/4-pi/2)(x-1)+(pi-1)(y-pi/2)#
#qquad +(y-pi/2)^2+(pi-1)(x-1)(y-pi/2)+... #
