So here we have this equation : xz+ylnx-x^2+4=0 and we took x as a function and others independent variables . I understood everything however i didn't know how to do the partial drivtes of x^2 and how they did that .

Let's differentiate both sides of the equation with respect to z, treating y as constant and x as a differentiable function of z. We get: (acz )+ az (y In x) (4) = 0 az ( 20 ? ) + Ox Oz 1 Ox Ox + y - 2x- +0 = 0 Oz az x Oz az Ox y ox 2x 0 Oz x Oz az Ox 2+ y 2ac = -X aC Ox zxty - 2x2 = - X Oz Ox Oz zacty - 2202