# Derivatives

## Derivative of a function $f$ at a number $a$ ### Notation

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

### Definition

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

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

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

#### As the slope of a tangent line

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

#### As a rate of change

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

## 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)