Verify normal derivative of a sphere.

I’ve search through many books and went online without any luck, I come up with a way to proof this and I need someone here to verify I am correct. Here is the problem:

Normal derivative is defined as:

Where is the unit outward normal of the surface of the sphere and for a small sphere with surface , the book gave:

The book claimed on a sphere:

. Where is the radius of the sphere.

I found the explanation from the PDE book of Strauss.

Where

I don’t get how to go from

This is the way I have to proof the identity:

In spherical coordinates:

We know and therefore

Where I substute n with r. But I still don't get the "-" sign yet.

Please give me your opinion. and help.

Thanks

Alan