fx y x2sinxySolutiondfdx 2xsinxy x2ycosxy Hence d2fdx2 2

Solution

df/dx = 2xsin(xy) + x²ycos(xy)
Hence d²f/dx² = 2sin(xy) + 2xycos(xy) + 2xycos(xy) - x²y²sin(xy) = (2-x²y²)sin(xy) + 4xycos(xy)

df/dy = x²*xcos(xy) = x³cos(xy)
Hence d²f/dy² = -x⁴sin(xy)

d²f/dxdy = 3x²cos(xy) - x³ysin(xy)

Hence the 2nd Order partial derivatives are:
d²f/dx² = (2-x²y²)sin(xy) + 4xycos(xy)
d²f/dy² = -x⁴sin(xy)
d²f/dxdy = 3x²cos(xy) - x³ysin(xy)