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

Diff as difference of two function values ${y}_{2}$ and ${y}_{1}$ with different values ${x}_{2}$ and ${x}_{1}$ of the independent variable.

While the difference quotient for sequences was given by the the difference of two terms with an index difference of $1$, i.e. ${A}_{n+1}-{A}_{n}$, the difference ${y}_{1}-{y}_{1}=f\left({x}_{2}\right)-f\left({x}_{1}\right)$ can be defined for an arbitrarily small difference $\Delta x={x}_{2}-{x}_{1}$ in the case of continuous functions.

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

$$\begin{array}{c}\text{differencequotient}:\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\frac{\Delta f}{\Delta x}=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}=\frac{f\left({x}_{1}+\Delta x\right)-f\left({x}_{1}\right)}{\Delta x};\hfill \\ \text{differentialquotient}:\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{\left(\frac{df}{dx}\right)}_{{x}_{1}}=\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\underset{\Delta x\to 0}{lim}{\left(\frac{\Delta f}{\Delta x}\right)}_{{x}_{1}}=\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\underset{\Delta x\to 0}{lim}\frac{f\left({x}_{1}+\Delta x\right)-f\left({x}_{1}\right)}{\Delta x}.\hfill \\ \hfill \end{array}$$

For $\Delta x>0$ we refer to a right-hand difference – or differential quotient, for $\Delta x<0$ 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 $x$, the first derivative of the function.

$${y}^{\prime}\left(x\right)={f}^{\prime}\left(x\right)=\frac{df}{dx}\left(x\right)\phantom{\rule{0em}{0ex}}=\phantom{\rule{0em}{0ex}}\underset{\Delta x\to 0}{limes}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}$$

As shorthand one writes the first derivative as ${y}^{\prime}$ (y-prime) or ${f}^{\prime}\left(x\right)$.

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:

$$df={f}^{\prime}\left(x\right)dx$$

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

$${y}^{\u2033}\left(x\right)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}={f}^{\u2033}\left(x\right)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}=\frac{d{f}^{\prime}}{dx}\left(x\right)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}.....\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}{y}^{\left(n\right)}\left(x\right)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}={f}^{\left(n\right)}\left(x\right)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}=\frac{d{y}^{\left(n-1\right)}}{dx}\left(x\right)\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}$$

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 $t$. For the derivative with respect to time the notation $\u1e8f$ (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 $t$). In this text we stick to the notation ${y}^{\prime}$, even if the independent variable is the time $t$.