tangent plane to surface at point

To find the plane, first get its normal vector.

normal vector to surface at point

The normal vector to surface $f$ is given by the gradient $\nabla f$. Evaluate the function at point to obtain the normal vector tangent to surface at point.

Link to original

Given the point and a normal vector, it is easy to make a plane.

plane from point and normal vector

Given a point $(x_0,y_0,z_0)$ and normal vector $\vec{n}$, a plane can be defined as follows:

$$\begin{array}{rl}{n}_x(x-x_0)+n_y(y-y_0)+n_z(z-z_0)& =0\\ n_x\cdot x+n_y\cdot y+n_z\cdot z& =n_x\cdot x_0+n_y\cdot y_0+n_z\cdot z_0\end{array}$$Link to original