First derivatives and tangents.

Let be the graph of a function defined on a neighborhood of . we denote by the point whose coordinates are . Through draw a line; suppose that this line intersects at another point . When gets arbitrarily close to , then the line sometimes gets closer to a limit position'', which will be called the tangent to at .

Actually , this is exactly what is yielded by the linear approxianmation of a function at a point (v.s. Definition 2.1):

 (6.1)

Denote , then the affine part of the right-hand expression in Equation (3) is . The equation is the equation of a line; this is te tangent to the graph of at the point whose coorsinate is equal to .

Theorem 6.3.1   If is differentiable at , then has a tangent at ; the slope of this tangent is equal to .

The equation of the tangent to at is

Example 6.3.2   Let . Then . If , we have and the equation of the tangent at the point is .

Noah Dana-Picard 2007-12-28