# Finding a Tangent Plane on a Surface Examples 3

Recall from the Finding a Tangent Plane on a Surface page that if $z = f(x, y)$ is a two variable real-valued function, then the equation of the tangent plane at a point $P(x_0, y_0)$ on the surface is given by the following formula:

(1)We will now look at some more examples of finding tangent planes to surfaces.

## Example 1

**Find the equation of the tangent plane on the paraboloid $f(x, y) = 3x^2 + 3y^2$ at the point $(1, 1, 6)$.**

We first compute the partial derivatives of $f$:

(2)Therefore we have that the equation of this tangent plane is given by:

(3)## Example 2

**Find the equation of the tangent plane on the hyperboloid $f(x, y) = \frac{1}{3}x^2 - y^2$ at the point $(3, 1, 2)$.**

We first compute the partial derivatives of $f$:

(4)Therefore the equation of this tangent plane is given by:

(5)