3432.sagews

var('x' 'y')
f(x,y)=x^2*y+y^2
fx=diff(f(x,y),x)
fy=diff(f(x,y),y)
fxx=diff(fx,x)
fxy=diff(fx,y)
fyx= diff(fy,x)
fyy= diff(fy,y)
print "1.fx:", fx
print "fy:", fy
print "2.fxx:", fxx
print "fyy:", fyy
print "fxy:", fxy
print "fyx:", fyx

implicit_plot3d((x-h)*fx+(y-k)*fy+f(h,k)==z,(x,-10,10),(y,-10,10),(z,-10,10))

1.fx: 2*x*y fy: x^2 + 2*y 2.fxx: 2*y fyy: 2 fxy: 2*x fyx: 2*x

var('x' 'y')
f(x,y)=e^x*cos(y)
fx=diff(f(x,y),x)
fy=diff(f(x,y),y)
fxx=diff(fx,x)
fxy=diff(fx,y)
fyx= diff(fy,x)
fyy= diff(fy,y)
print "3.fx:", fx
print "fy:", fy
print "fxx:", fxx
print "fyy:", fyy
print "fxy:", fxy
print "fyx:", fyx
h=0
k=0
var('z')

a=implicit_plot3d((x-h)*fx+(y-k)*fy+f(h,k)==z,(x,-10,10),(y,-10,10),(z,-10,10))
b=implicit_plot3d(f(x,y)==z,(x,-10,10),(y,-10,10),(z,-10,10))
a+b
z=(.1*e^.1*sin(.1)+.1*e^.1*cos(.1)+f(h,k))

print "z:",z

3.fx: e^x*cos(y) fy: -e^x*sin(y) fxx: e^x*cos(y) fyy: -e^x*cos(y) fxy: -e^x*sin(y) fyx: -e^x*sin(y) z: 1.12099826555596