calculus

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

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  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

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