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In: Economics

Write down expressions for the first-order partial derivatives, ?z and ?z for ?x ?y (a)z=x2 +4y5...

Write down expressions for the first-order partial derivatives, ?z and ?z for ?x ?y (a)z=x2 +4y5 (b)z=3x3 ?2ey (c)z=xy+6y (d)z=x6y2 +5y3

Solutions

Expert Solution

a.

z = x^2 + 4y^5

(dz/dx) = Partial derivative of z with respect to x

             = 2x ^(2 – 1) + 0

             = 2x [Answer]

z = x^2 + 4y^5

(dz/dy) = Partial derivative of z with respect to y

             = 0 + (4 × 5) y ^(5 – 1)

             = 20y^4 [Answer]

c.

z = xy + 6y

(dz/dx) = Partial derivative of z with respect to x

           = y × x ^(1 – 1) + 0

             = y × x ^0

             = y × 1

             = y [Answer]

z = xy + 6y

(dz/dy) = Partial derivative of z with respect to y

           = x × y^(1 – 1) + 6y^(1 -1)

             = x × y^0 + 6y^0

             = x × 1 + 6 × 1

             = x + 6 [Answer]

d.

z = x6y^2 + 5y^3

(dz/dx) = Partial derivative of z with respect to x

             = (6y^2) × x ^(1 – 1) + 0

             = (6y^2) × x ^0

             = (6y^2) × 1

             = 6y^2 [Answer]

z = x6y^2 + 5y^3

(dz/dy) = Partial derivative of z with respect to y

             = x(6 × 2)y^(2 -1) + (5 × 3)y^(3 – 1)

             = 12xy + 15y^2 [Answer]

Rahul Sunny answered 2 years ago
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