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 = x8y9, x = s cos(t), y = s sin(t)
∂z/∂s =
∂z/∂t =

Step-by-step explanation:

Using chain rule to find the partial deriviative of z with respect to s and t i.e ∂z/∂s and ∂z/∂t, we will use the following formula since it is composite in nature;

∂z/∂s = ∂z/∂x*∂x/∂s +  ∂z/∂y*∂y/∂s

Given the following relationships z = x⁸y⁹, x = s cos(t), y = s sin(t)

∂z/∂x = 8x⁷y⁹, ∂x/∂s = cos(t), ∂z/∂y = 9x⁸y⁸ and ∂y/∂s = sin(t)

On substitution;

∂z/∂s = 8x⁷y⁹(cos(t)) + 9x⁸y⁸ sin(t)

∂z/∂s = 8(scost)⁷(s sint)⁹(cos(t)) + 9(s cost)⁸(s sint)⁸ sin(t)

∂z/∂s = (8s⁷cos⁸t)s⁹sin⁹t + (9s⁸cos⁸t)s⁸sin⁹t

∂z/∂s = 8s¹⁶cos⁸tsin⁹t + 9s¹⁶cos⁸tsin⁹t

∂z/∂s = 17s¹⁶cos⁸tsin⁹t

∂z/∂t =  ∂z/∂x*∂x/∂t +  ∂z/∂y*∂y/∂t

∂x/∂t = -s sin(t) and ∂y/∂t = s cos(t)

∂z/∂t  = 8x⁷y⁹*(-s sint) + 9x⁸y⁸* (s cos(t))

∂z/∂t = 8(scost)⁷(s sint)⁹(-s sint) + 9(s cost)⁸(s sint)⁸(s cos(t))

∂z/∂t = -8s¹⁷cos⁷tsin¹⁰t + 9s¹⁷cos⁹tsin⁸t

∂z/∂t = -s¹⁷cos⁷tsin⁸t(8sin²t-9cos²t)

-60.75

step-by-step explanation:

just add the numbers then put the negative sign on it