### 5.2 Difference Quotient and Differential Quotient

For a continuous function the variable $Math content$ can have any value within its domain of definition $Math content$. Thus as for sequences one can define a difference quotient

Diff as difference of two function values $Math content$ and $Math content$ with different values $Math content$ and $Math content$ of the independent variable.

While the difference quotient for sequences was given by the the difference of two terms with an index difference of $Math content$, i.e. $Math content$, the difference $Math content$ can be defined for an arbitrarily small difference $Math content$ in the case of continuous functions.

In addition one can define a differential quotient as limit for an infinitesimal distance of the variables $Math content$. Thus it becomes a local property of the function in every point $Math content$, in which such a unique value exists, i.e. at which the function is differentiable.

$Math content$

For $Math content$ we refer to a right-hand difference – or differential quotient, for $Math content$ to a left-hand one. If both differential quotients exist and are equal, then the function is uniquely differentiable at this point.

If the function is uniquely differentiable in every point of its domain of definition – it is then also continuous – its differential quotient is a continuous function of the variable $Math content$, the first derivative of the function.

$Math content$

As shorthand one writes the first derivative as $Math content$ (y-prime) or $Math content$.

If the differential quotient exists, it is a proper ration of two number as their respective limits. Thus one can treat both denominator and numerator as such:

$Math content$

If the first derivative is uniquely differentiable in every point of the domain of definition one can define the second derivative, an so on:

$Math content$

Of particular practical importance are functions that can arbitrarily often continuously differentiated also called “smooth” , such as the trigonometric functions.

In physics the independent variable is often the time $Math content$. For the derivative with respect to time the notation $Math content$ (y-dot) has been adopted in hand writing and printing. This is somewhat unfortunate for our purposes, since this sign cannot be directly entered on the keyboard of a PC, and also one cannot enter it as a character with two meanings ( y and derivative with respect to $Math content$). In this text we stick to the notation $Math content$, even if the independent variable is the time $Math content$.