Use the chain rule to find and where z exs tan y x 5s 3t

Use the chain rule to find and , where z = exs tan y, x = 5s + 3t, y = 6s / 6t First the pieces: And putting it all together:

Solution

dz/dx = ye^(xy)tany dz/dy = xe^(xy)tany + e^(xy)sec^2y dx/ds = 5 dx/dt = 3 dy/ds = 1/t dy/dt = -s/t^2 dz/ds = dz/dx*dx/ds + dz/dy*dz/ds dz/ds = 5(ye^(xy)tany) + 1/t[xe^(xy)tany + e^(xy)sec^2y] dz/ds = 5yz + 1/t[xz + e^(xy)sec^2y] dz/dt = dz/dx*dx/dt + dz/dy*dz/dt dz/dt = 3(ye^(xy)tany) - s/t^2[xe^(xy)tany + e^(xy)sec^2y] dz/dt = 3yz - s/t^2[xz + e^(xy)sec^2y]