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

giorgiana1976 | Student

To get the second partial derivatives, we must determine the f1st partial derivative of the given function.

We'll differentiate with respect to x: 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