More practice problems
Use the chain rule to find ∂Y/∂X for the following:
Y = (3X2 - 13)3
Chain Rule ∂Y/∂X = ∂Y/∂Z* ∂Z/∂X let Z = 3X2 - 13
Y = Z3
∂Y/∂Z = 3Z2
∂Z/∂X = 6X
∂Y/∂X = 3Z2 *6X = 18X(3X2 - 13)2
Y = (8X3 - 5X2)9
Chain Rule ∂Y/∂X = ∂Y/∂Z* ∂Z/∂X let Z = 8X3 - 5X2
Y = Z9
∂Y/∂Z =9Z8
∂Z/∂X = 24X2 - 10X
∂Y/∂X = 9( 8X3 - 5X2 )8 (24X2 - 10X) = (216X2 - 90X)( 8X3 - 5X2 )8
Y = (aX2 + b)4
Chain Rule ∂Y/∂X = ∂Y/∂Z* ∂Z/∂X let Z = aX2 + b
Y = Z4
∂Y/∂Z = 4Z3
∂Z/∂X = 2aX
∂Y/∂X =4( aX2 + b)3 2aX = 8aX ( aX2 + b)3
Use the product rule to find ∂Y/∂X for the following:
Y = (9X2 - 2)(3X + 1)
f (X) = (9X2 - 2)
Product Rule Y = f (X) * g ' (X) + f ' (X) * g(X)
f ' (X) = 18X
g(X) = (3X + 1)
g ' (X) = 3
∂Y/∂X = (9X2 - 2)3 + (3X + 1)18X = 27X2 - 6 + 54X2 + 18X
∂Y/∂X = 81X2 + 18X - 6
Y = (aX2 - b)(cX3)
f (X) = (aX2 - b)
Product Rule Y = f (X) * g ' (X) + f ' (X) * g(X)
f ' (X) = 2aX
g(X) = (cX3)
g ' (X) = 3cX2
∂Y/∂X = (aX2 - b) 3cX2 + 2aX (cX3) = 3acX4 - 3bcX2 + 2acX4 = 5acX4 - 3bcX2
Y = (3X + 11)(6X2 - 5X)
Product Rule Y = f (X) * g ' (X) + f ' (X) * g(X)
f (X) = (3X + 11)
g(X) = (6X2 - 5X)
f ' (X) = 3
g ' (X) = 12X - 5
∂Y/∂X = (3X + 11) (12X - 5) + 3 (6X2 - 5X) = (3X + 11) (12X - 5) + (18X2 - 15X)
Use the quotient rule to find ∂Y/∂X for the following:
Y = ( X2 + 3) / X
f (X) = ( X2 + 3)
∂Y / ∂X = {f ' (X) g (X) - f (X) g ' (X) } / g 2(X)
f ' (X) = 2X
g (X) = X
g ' (X) = 1
∂Y/∂X = {2X X - ( X2 + 3)1}/X2 = {2X2 - X2 - 3}/X2 = {X2 - 3}/X2 = 1 - 3X-2
∂Y / ∂X = {f ' (X) g (X) - f (X) g ' (X) } / g 2(X)
Y = 4X / (X + 5)
f (X) = 4X
f ' (X) = 4
g (X) = (X + 5)
∂Y/∂X = {4 (X + 5) - 4X1}/(X + 5)2 = {4X + 20 - 4X} / (X + 5)2 =
g ' (X) = 1
20 /(X + 5)2
Y = (7X4 + 5X3) / (4X5 - 3X2)
∂Y / ∂X = {f ' (X) g (X) - f (X) g ' (X) } / g 2(X)
f (X) = 7X4+ 5X3
g (X) = (4X5 - 3X2)
f ' (X) = 28X3
+15X2
g ' (X) = (20X4 - 6X)
∂Y/∂X = {(28X3 +15X2) (4X5 - 3X2) - (7X4+ 5X3)(20X4 - 6X)} / (4X5 - 3X2)2
Use the concept of a partial differential to find ∂Y/∂X1 and ∂Y/∂X2 for the following:
Y = 2X13 - 11X12X2 + 3X22
∂Y/∂X1 = 6 X12 - 22X1X2
∂Y/∂X2 = - 11X12 + 6X2
Y = 7X1 + 5X1X22 - 9X23
∂Y/∂X1 = 7 + 5X22
∂Y/∂X2 = 10X1X2 - 27X22
Y = (3X1 - 2X2) / (X12 + 3X2)
∂Y / ∂X = {f ' (X) g (X) - f (X) g ' (X) } / g 2(X)
∂Y/∂X1 =
f (X) = (3X1 - 2X2)
f ' (X) = 3
g (X) = (X12 + 3X2)
g ' (X) =2X1
∂Y/∂X1 = {3(X12 + 3X2) - 2X1(3X1 - 2X2)} /(X12 + 3X2)2
∂Y/∂X1 = (3X12 + 9X2 - 6X12 + 4X1X2) / (X12 + 3X2)2=(-3X12 + 9X2.+4X1X2) / (X12 + 3X2)2
∂Y/∂X2 =
f (X) = (3X1 - 2X2)
f ' (X) = -2
g (X) = (X12 + 3X2)
g ' (X) = 3
∂Y/∂X2 = {-2 (X12 + 3X2) - 3 (3X1 - 2X2)} / (X12 + 3X2)2
∂Y/∂X2 = (-2X12 - 6X2 - 9X1 + 6X2) / (X12 + 3X2)2 = (-2X12 - 9X1) / (X12 + 3X2)2