Use the Chain Rule to find zs and zt Enter your answer only

Use the Chain Rule to find z/s and z/t. (Enter your answer only in terms of s and t. Please use * for multiplication between all factors.) z = x2y7, x = s cos(t), y = s sin(t)

Solution

Use the Chain Rule to find z/s and z/t.

Z=x²y⁷  , x= s cost , y = s sint

# z/s = (z/x)(x/s) + (z/y)(y/s)

z/s = 2xy⁷(cost) + 7x²y⁶ (sint) = 2(scost)(ssint)⁷ (cost)+ 7(scost)²(ssint)⁶(sint)

z/s = 2s⁸ cos²t sin⁷t + 7s⁸ cos²t sin⁷t = 9s⁸ cos²t sin⁷t

# z/t = (z/x)(x/t) + (z/y)(y/t)

z/t = 2xy⁷ (-s sint) + 7x²y⁶ (s cost) = 2(scost)(ssint)⁷ (-s sint)+ 7(scost)²(ssint)⁶(s cost)

z/t = -2s⁹ cost sin⁸t + 7s⁹ cos³t sin⁶t = s⁹ cost sin⁶t (7cos²t - 2sin²t)