I've been taking a multivariable calculus course recently and we just talked about tangent planes to surfaces. As it was brought up in the course, the tangent plane of a surface is found by taking the partial derivatives with respect to x and y and turning them into the vectors:
<1, 0, fx> and <0, 1, fy> where fx and fy are the partial derivatives.
Then, the cross product of these two vectors produces a normal to the surface at that point and you can then use it to create a plane by doing the dot product with the vector: <x-a, y-b, z-c> to get the equation of your plane.
I understand that this is kind of a long process that can be shortened by simply remembering the formula for the tangent plane but I wanted to describe how I was thinking about getting the tangent plane.
My problem is that this only seems to work with 3-Dimensional surfaces (ie. f(x,y)). We were recently tasked with finding the tangent plane of a sphere at a certain point, but this method described above doesn't seem to work. Is this because the equation of a sphere depends on x, y, and z (ie. f(x,y,z)), or am I thinking about this the wrong way.
I saw some places online talking about having to use the gradient of the function, but my textbook hasn't brought this up yet. My textbook has also only shown examples of tangent planes for f(x,y) functions.
Does my method above only work for f(x,y) functions?
I might be thinking about this all wrong. Any help would be greatly appreciated.