# Derivatives

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## Derivative of a function at a number [edit | edit source]

### Notation[edit | edit source]

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

### Definition[edit | edit source]

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 :

### Interpretations[edit | edit source]

#### As the slope of a tangent line[edit | edit source]

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

#### As a rate of change[edit | edit source]

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 curve[edit | edit source]

### Vocabulary[edit | edit source]

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

### Definition[edit | edit source]

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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