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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