z=e^(xy^2)z=exy2
dz/dt= (del z)/(delx) dx/dt + (delz)/(dely) dy/dtdzdt=∂z∂xdxdt+∂z∂ydydt
=e^(xy^2) (y^2) (cost -t sint) +e^(xy^2) (x) (2y dy/dt) ( sint +t cost)=exy2(y2)(cost−tsint)+exy2(x)(2ydydt)(sint+tcost)
=e^(xy^2) (y^2) (cost - tsin t)+e^(xy^2) 2xy (sint +tcost)^2=exy2(y2)(cost−tsint)+exy22xy(sint+tcost)2
At t=pi/2, cost =0, sint =1, x=0 and y= pi/2t=π2,cost=0,sint=1,x=0andy=π2
Therefore dz/dt =(pi/2)^2 (-pi/2)= -(pi/2)^3dzdt=(π2)2(−π2)=−(π2)3