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)

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