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Concept of a derivative

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Let be a fixed point and be a variable point on the curve as shown on about figure. Then the slope of the line AP is given by or . When the variable point P moves closer and closer to A along the curve , i.e. . the line AP becomes the tangent line of the curve at the point A. Hence, the slope of the tangent line at the point A is equal to . This term is defined to be the derivative of at and is usually denoted by . The definition of derivative at any point x may be defined as follows.

 

Definition 1. Let be a function defined on the interval and .

is said to be differentiable at (or have a derivative at ) if the limit exists. This lime value is denoted by or and is called the derivative of at .

If has a derivative at every point x in , then is said to be differentiable on .

Remark. As , the difference between x and is very small, i.e. tends to zero.

Usually, this difference is denoted by h or . Then the derivative at may be rewritten as

. (First Principle)

 


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