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A tangent plane is a plane that touches the surface of a curve defined in 3-space at a given point P. The normal vector of the tangent plane is perpendicular to the surface at the point P. The surface must be "smooth" near the point P.

Given a three dimensional surface, it can be useful to find the plane tangent to a certain point on the surface. This can be easily accomplished using some basic calculus.

If the function `z = f(x,y)` is smooth and has both x and y partial derivatives at the point `(a,b)`, then its tangent plane can be found. The plane tangent to the surface `z = f(x,y)` at the point `P(a,b,f(a, b))` contains the lines that are tangent to both of the following curves:

```z = f(x,b) holding constant y = b
z = f(a,y) holding constant x = a
```

To solve for a the general plane equation, we need the definition of a plane:

```A(x-a) + B(y-b) + C(z-c) = 0
z–c = -A/C(x-a) + -B/C(y-b)
if we set p = -A/C and q = -B/C then
z-c = p(x-a) + q(y-b)
```

Going back to the previous statement, we know that the plane has to contain both tangent lines to the respect x and y curves. When `z = f(x,b)` then the equation simplifies into `z-c = p(x-a)`, or the equation of a line. To find the slope of that line, a partial derivative with respect to x is done. That gives us `p =∂f/∂x` and `q =∂f/∂y`. The last part of the equation is `c`, but it's simple to see that when given the point `P(a,b,f(a,b))`, then `c = f(a,b)`. Finally, we get the general tangent plane equation for `z = f(x,y)` at the point `P(a,b,f(x,y))`:

```z - f(a,b) = =∂f/∂x(x-a) + =∂f/∂y(y-b)
```
Sources: Calculus with analytic geometry, by Edwards and Penny, Fifth edition.

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