partial derivative w.r.t x (52-3x^2-4y^2)^0.5

Let's call this function z

So

z = (52 - 3x^2 - 4y^2)^(1/2)  .... so.....

z_x = (1/2)(52 - 3x^2 - 4y^2)^(-1/2) * ∂/∂x (52 - 3x^2 - 4y^2) =

(1/2) (52 - 3x^2 - 4y^2)^(-1/2) * (-6x) =

(-3x)(52 - 3x^2 - 4y^2)^(-1/2)

Remember....we're differetiating with respect to x, so any "y" terms are treated as constants when taking their derivatives.....

(52 - 3x² -4y²)^1/2

First derivative --->   (1/2)(52 - 3x² - 4y²)^-1/2(-6x - 8yy')