# How to compute the normal to the ellipsoid at the point on the surface of ellipsoid?

Let the equation of an ellipsoid be:

$$2x^2+y^2+2z^2=5$$

And the point on the surface of ellipsoid be:

$$(1,1,1)$$

How to compute the normal to the ellipsoid at the point on the surface of ellipsoid?

I read some article says that I can compute it by gradient, but I am not sure how to do it...

If $c$ is a regular value of a smooth function $f$, then $S=f^{-1}(c)$ is a surface for which $\nabla f(p)$ is normal to $T_pS$. Here $f(x,y,z) = 2x^2+y^2+2z^2$ and $p = (1,1,1)$, so the normal to the ellipsoid at $p$ is $$\nabla f(1,1,1) = (4x,2y,4z)\big|_{x=y=z=1} = (4,2,4).$$