Use the Chain Rule to find the indicated partial derivatives.

Question

z = x^4 + x^2y, x=s+2t −u, y =stu^2; ∂z/∂s ∂z/∂t ∂z/∂u when s = 3, t = 4, u = 5?

Answer 1

(1).

The function are z = x⁴ + x²y, x = s+2t-u, y = stu², and the values are s = 3, t = 4, u = 5.

Substitute x = s+2t-u and y = stu² in the function z = x⁴ + x²y.

z = (s+2t-u)⁴ + (s+2t-u)²(stu²)

Partial differentiate with respect to s to the above function.

∂z/∂s = (∂/∂s)[(s+2t-u)⁴ + (s+2t-u)²(stu²)]

∂z/∂s = 4(s+2t-u)³ + [(s+2t-u)²(tu²) + 2(s+2t-u)(stu²)]

Substitute s = 3, t = 4 and u = 5 in the above function.

∂z/∂s = 4[3+2(4)-5]³ + [3+2(4)-5]²(4 * 5²) + 2[3+2(4)-5](3 * 4 * 5²)

∂z/∂s = 864 + 3600 + 3600

∂z/∂s = 8064

Answer 2

(2).

The function are z = x⁴ + x²y, x = s+2t-u, y = stu², and the values are s = 3, t = 4, u = 5.

Substitute x = s+2t-u and y = stu² in the function z = x⁴ + x²y.

z = (s+2t-u)⁴ + (s+2t-u)²(stu²)

Partial differentiate with respect to t to the above function.

∂z/∂t = (∂/∂t)[(s+2t-u)⁴ + (s+2t-u)²(stu²)]

∂z/∂t = 8(s+2t-u)³ + [(s+2t-u)²(su²) + 4(s+2t-u)(stu²)]

Substitute s = 3, t = 4 and u = 5 in the above function.

∂z/∂t = 8[3+2(4)-5]³ + [3+2(4)-5]²(3 * 5²) + 4[3+2(4)-5](3 * 4 * 5²)

∂z/∂t = 1728 + 2700 + 7200

∂z/∂t = 11628.

Answer 3

(3).

The function are z = x⁴ + x²y, x = s+2t-u, y = stu², and the values are s = 3, t = 4, u = 5.

Substitute x = s+2t-u and y = stu² in the function z = x⁴ + x²y.

z = (s+2t-u)⁴ + (s+2t-u)²(stu²)

Partial differentiate with respect to u to the above function.

∂z/∂u = (∂/∂u)[(s+2t-u)⁴ + (s+2t-u)²(stu²)]

∂z/∂u = -4(s+2t-u)³ + [2(s+2t-u)²(stu) - 2(s+2t-u)(stu²)]

Substitute s = 3, t = 4 and u = 5 in the above function.

∂z/∂u = -4[3+2(4)-5]³ + 2[3+2(4)-5]²(3 * 4 * 5) - 2[3+2(4)-5](3 * 4 * 5²)

∂z/∂u = - 864 + 4320 - 3600

∂z/∂u = -144.