Let z=(x^2+y^2)^(3/2).

Question

Let z=(x^2+y^2)^(3/2). What is (d^2*z)/(dx*dy) at (sqrt(2), sqrt(2))?

dz/dy=3y(x^2+y^2)^(1/2)

d/dx=3x(x^2+y^2)^(1/2)

What should I do next?

Answer 1

∂z/∂y = 3y(x^2+y^2)^(1/2)

∂²z/∂y∂x = (3y/2)2x/(x^2+y^2)^(1/2) = 3xy/(x^2+y^2)^(1/2) = 3 at (sqrt(2), sqrt(2))