How do I show that the direction of second differential vector is normal to the surface?
The position vector is a function of two variables. The "second differential" with respect to which variable? If, for example, we take, as parametric equations for the unit sphere, then and then the mixed derivative is and the dot product of that with is so the mixed second derivative is perpendicular to the surface.
But the second derivative with respect to both times is and that is NOT perpendicular to the surface.