Find all first and second partial derivatives of z with respect to x and y if x^2+4y^2+16z^2−64=0.

Answer 1

x^2+4y^2+16z^2−64 = 0
16z^2 = 64-x^2-4y^2
2z ∂z/∂x = -2x
2z ∂z/dy = -8y

so,
∂z/∂x = -x/z
∂z/∂y = -4y/z

∂^2z/∂x^2 = -(x^2+z^2)/z^3

and you can do the others similarly, using the quotient rule or the product rule

Answer 2

compute the first order partial derivatives of f(x,y) with respect to x and y. assume a constant