What are the second partial derivatives , z (x,y) and z (y,x) if z = x^2*y + 2x*e^1/y ?

To determine the second partial derivatives, we'll have to
calculate first partial derivative for given
expression.

We'll calculate
dz/dx.

z=x^2*y +
2x*e^1/y

We'll differentiate the expression of z with
respect to x, treating y as a constant.

dz/dx =
(d/dx)(x^2*y + 2x*e^1/y)

dz/dx = y(d/dx)(x^2) +
(2e^1/y)(d/dx)(x)

The first partial derivative, with
respect to x, is:

dz/dx = 2xy +
2e^1/y

d^2z/dx*dy= x^2 -
2xe^1/y/y^2

To calculate the first partial derivative of z,
with respect to y, we'll have:

dz/dy = (d/dy)(x^2*y +
2x*e^1/y)

dz/dy = x^2 -
2xe^1/y/y^2

d^2z/dy*dx = 2x -
2e^1/y/y^2