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 ${\displaystyle x=a}$:

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)