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Concept of a derivative
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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