Problem finding normal line equation using partial derivative and gradient vector

I'm trying to solve a problem but the answer is different from the book.

So wondering what am i doing wrong. The problem is like this:

Find equations for the lines that are normal to function:

at

The way i did was first find the gradient vector:

Then get the derivative which is(because we know that slope

of a function is it's derivative at that point):

Finally the equation of normal line is:

Is this right or am i wrong? The answer back of my book is

Why book's answer is different? Any ideas?