is this correct, second-order partial derivatives

arslan

Find the first- and second-order derivatives of the following functions:

y=2x^3+4x^2-x+5

y=(2x+1)(4x-2)

y=(x^2+2)^2

Find all the first-order and second-order partial derivatives of the following functions:

z=3x^2-2y^4

z=2xy^2+0.5x^2

z=y^3+2x^2 y^2-4x+2

this is what i have been able to do i just wanted someone to check if its correct

y=2x^3+4x^2-x+5
dy/dx = 6x^2+8x-1

y=(2x+1)(4x-2)
dy/dx = (2) (4x-2)+(2x+1)(4)
dy/dx = 8x-4+8x+4
dy/dx=16x

y=(x^2+2)^2
dy/dx = 2(x^2+2) (2x)
dy/dx = (2x^2+4)(2x)
dy/dx = 4x^3+8x

z=3x^2-2y^4
∂z/∂x = 6x
∂^z/∂x^2 = 6

∂z/∂y = -8y^3
∂^2z/∂y^2 = -8(3y^2) = -24y^2

z=2xy^2+0.5x^2
∂z/∂x = 2y^2+0.5(2x) = 2y^2+x
∂^2z/∂x^2 = 1
∂z/∂y = 2x(2y) = 4xy
∂^2z/∂y^2 = 4x

z=y^3+2x^2 y^2-4x+2
∂z/∂x = 2y^2 (2x) - 4 = 4xy^2-4
∂^2z/∂x^2 =4y^2
∂z/∂y = 3y^2+2x^2(2y) = 3y^2+4x^2y
∂^2z/∂y^2 = 3(2y)+4x^2 = 6y+4x^2

yeKciM

lol it's ok
but for simple things u can download "microsoft math 3.0." so u can check ur work (Wink)

arslan

Ackbeet

MHF Hall of Honor
For completeness, you should compute the mixed partial derivatives, such as $$\displaystyle \frac{\partial^{2}z}{\partial x\,\partial y},$$ etc.