I quote a paragraph of "Morse Theory and floer homology"
The critical points of the height function on the sphere are nondegenerate. Indeed, in the neighborhood of the point $(0, 0, \varepsilon) \in\Bbb S^2$, we can take $(x, y)$ as local coordinates. Then $$f(x, y) = z = \varepsilon\sqrt{1 − x^2 − y^2},$$ is a function whose second-order derivative is the quadratic form $$(d^2f)(0,0,\varepsilon)(x, y) = −\varepsilon(x^2 + y^2),$$ which is indeed nondegenerate.
Question : How to calculate second -order derivative of this height function?
If $x_0$ is a critical point, you can compute the Hessian in local coordinates as you would do with a function defined on an open set of $\mathbb{R}^n$ so $Hess (f)_{x_0} = (\frac {\partial f(x_0)}{\partial x_i \partial x_j})_{i,j}$. In your case since $f$ is represented in local coordinates near the critical point $(0,0,\epsilon)$ as $f(x,y)= \epsilon\sqrt{1-x^2-y^2}$, you simply have to compute the partial derivatives $\frac {\partial{f}(x,y)}{\partial x_i \partial x_j}$ and evaluate them at $(x,y)=(0,0)$. You will get a diagonal matrix $diag(-\epsilon,-\epsilon)$ so the quadratic form associated is $(x,y)\mapsto -\epsilon(x^2+y^2)$.