Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag

Mathematics-Online lexicon:

Derivative


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

A function $ f$ is differentiable at the point $ a$ if the limit

$\displaystyle f^\prime(a)= \lim_{h\to0} \frac{f(a+h)-f(a)}{h} $

exists. This limit is called the derivative of $ f$ at $ a$.

\includegraphics[width=0.6\linewidth]{a_ableitung_bild_1}

Geometrically, differentiability means that the slopes of the secants converge to the slope of the tangent given by

$\displaystyle y=f(a)+f^\prime(a)(x-a)
\,.
$

We also write

$\displaystyle f^{\prime}(x)=\frac{d}{dx}f(x)=\frac{dy}{dx}
$

with $ y=f(x)$. This notation symbolizes the limit $ \Delta x\to0$ for the difference quotient.

Higher derivatives are denoted by $ f^{\prime\prime},f^{\prime\prime\prime},\ldots$ or $ f^{(2)},f^{(3)},\ldots$, respectively.

We say that a function $ f$ is differentiable on a set $ D$ if $ f^\prime(x)$ exists for all $ x\in D$.


[Examples] [Links]

  automatically generated 6/14/2016