# Derivatives

## Derivative of a function ${\displaystyle f}$ at a number ${\displaystyle a}$

### Notation

We denote the derivative of a function ${\displaystyle f}$  at a number ${\displaystyle a}$  as ${\displaystyle f'(a)\,\!}$ .

### Definition

The derivative of a function ${\displaystyle f}$  at a number ${\displaystyle a}$  a is given by the following limit (if it exists):

${\displaystyle f'(a)=\lim _{h\rightarrow 0}{\frac {f(a+h)-f(a)}{h}}}$

An analagous equation can be defined by letting ${\displaystyle x=(a+h)}$ . Then ${\displaystyle h=(x-a)}$ , which shows that when ${\displaystyle x}$  approaches ${\displaystyle a}$ , ${\displaystyle h}$  approaches ${\displaystyle 0}$ :

${\displaystyle f'(a)=\lim _{x\rightarrow a}{\frac {f(x)-f(a)}{x-a}}}$

### Interpretations

#### As the slope of a tangent line

Given a function ${\displaystyle y=f(x)\,\!}$ , the derivative ${\displaystyle f'(a)\,\!}$  can be understood as the slope of the tangent line to ${\displaystyle f(x)}$  at ${\displaystyle x=a}$ :

#### As a rate of change

The derivative of a function ${\displaystyle f(x)}$  at a number ${\displaystyle a}$  can be understood as the instantaneous rate of change of ${\displaystyle f(x)}$  when ${\displaystyle x=a}$ .

## At a tangent to one point of a curve

### Vocabulary

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

### Definition

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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