Derivatives

Derivative of a function at a number Edit

NotationEdit

We denote the derivative of a function   at a number   as  .

DefinitionEdit

The derivative of a function   at a number   a is given by the following limit (if it exists):

 


An analagous equation can be defined by letting  . Then  , which shows that when   approaches  ,   approaches  :

 

InterpretationsEdit

As the slope of a tangent lineEdit

Given a function  , the derivative   can be understood as the slope of the tangent line to   at  :

As a rate of changeEdit

The derivative of a function   at a number   can be understood as the instantaneous rate of change of   when  .

At a tangent to one point of a curveEdit

VocabularyEdit

The point A(a ; f(a)) is the point in contact of the tangent and Cf.

DefinitionEdit

If f is differentiable in a, then the curve C admits at a point A which has for coordinates (a ; f(a)), a tangent : it is the straight line passing by A and of direction coefficient f'(a). An equation of that tangent is written: y = f'(a)*(x-a)+f(a)


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