Does there exist any particular way of finding the centre of an ellipse by partially differentiating the equation with respect to x and y both and then finding the point of intersection??

I'm asking this because I know it works for hyperbolas and due to symmetry in their relations I would like to know whether it works for ellipses too...

This should work according to me.

If you get the equation of the normals at the various points, two would be enough, you should get your centre of elipse for a regular elipse with major axis and minor axis parallel to the x or y axes.

If you have a 'slant' elipse, you can differentiate for gradient = 0 and infinite to get the points you need, then you get the line passing through them and their intersection will be the centre.