Tangent Plane to a Level Surface

If given a level surface () denoted as , and a point on the surface , then the tangent plane at this point is a plane passing through the point with the normal vector parallel to . Where is the gradient function at point .

Ok, lets try that again...
We have a function and a point . Want an equation of a plane tangent to it (see equation of a plane):

Which means we need two things:

1. A vector normal to our plane.
2. A point on the plane.

We can find 1. A vector normal to our plane by finding the gradient function of , our normal vector is any vector parallel to this gradient function at point , or in other terms a scalar multiple of the gradient function at point .

We can find 2. A point on the plane by simply using the provided point, .

See Multivariable Calculus Notes - Chapter 14 - Sections 5-6 for more info.