7.3. Differentiable Functions

Again we take the result of 7.1. as our guideline. We have already transferred the spectrum of secant gradients into a mathematical notion, so the remaining task will be to deal with the tangent gradient. We take [7.1.2] as a pattern.

Definition:  Let $a\in A$ be an accumulation point of $A\subset ℝ$.   We call a function  $f:A\to ℝ$  differentiable at a, if the difference quotient function  ${\displaystyle m_a}$ is continuously extendable at a. In this case the real number (read:  f-prime at a) ${\displaystyle f^{\prime }(a)≔\underset{x\to a}{\lim }{m}_a(x)=\underset{x\to a}{\lim }\frac{f(x)-f(a)}{x-a}}$ [7.3.1] is called derivative (more precise: derivative number) of  f at a.

Consider:

• Being the limit of secant gradients we regard  $f^{\prime }(a)$ as the gradient of the tangent at  f  in (a, f(a)).

• As a is a accumulation point of A - and thus also of $A\backslash \left\{a\right\}$ -   ${\displaystyle m_a}$ has at most one continuous extension at a. Therefor the limit at a, i.e. the number  $f^{\prime }(a)$ is uniquely determined.

• The symbol  ${\displaystyle f^{\prime }(a)}$ is due to Cauchy. Although quite common it is not seldom replaced by Leibniz's notation ${\displaystyle \frac{df}{dx}(a)}$ and ${\displaystyle \frac{d}{dx}f(a)}$ respectively. The idea behind is, that when "performing the limit process" the quotient of the differences ${\displaystyle \mathrm{\Delta }f=f(x)-f(a)}$ and ${\displaystyle \mathrm{\Delta }x=x-a}$ will switch over into the mystic

  i

  Note that there was no precise limit idea for Leibniz (1646 - 1716) and it was not until Cauchy (1789 - 1857) who developed a modern, satisfactory concept.

  quotient of differentials  df and dx. We read ${\displaystyle \frac{df}{dx}}$ as "dee eff dee ecks" to indicate, at least when reading, that there is no real quotient.

• Again physics has its own notation at this point: With time related functions, e.g. with those of the ${\displaystyle t\mapsto s(t)}$ type (cf. the note in [7.2]) the derivative number is denoted by a dot symbol

  ${\displaystyle \dot{s}(t_0)=\underset{t\to t_0}{\lim }\frac{\mathrm{\Delta }s}{\mathrm{\Delta }t}=\underset{t\to t_0}{\lim }\frac{s(t)-s(t_0)}{t-t_0}}$

  and referred to as the instantaneous speed at the time ${\displaystyle t_0}$. It is usually replaced by  ${\displaystyle v(t_0)=\dot{s}(t_0)}$.

  It is this textual concept that the alternative term "instantaneous rate of change of the function  f at a" for the derivative  ${\displaystyle f^{\prime }(a)}$ comes from.

  
   

Example:   Every linear function  $mX+b$  is differentiable at each $a\in ℝ$ with   ►   $(mX+b{)}^{\prime }(a)=m$ [7.3.2] Proof: According [7.2.2]  ${\displaystyle m_a}$ is continuously extendable at a by the constant function m on $ℝ$. Thus we have:   $(mX+b{)}^{\prime }(a)=\underset{x\to a}{\lim }{m}_a(x)=m(a)=m$.   With $n\in \mathbb{N}$ the power function  $X^n$  is differentiable at every $a\in ℝ$ and   ►   $(X^n{)}^{\prime }(a)=n{a}^{n-1}$ [7.3.3] Proof: From [7.2.3] we see that  ${\displaystyle m_a}$ is continuously extendable at a by ${\displaystyle \sum _{i=0}^{n-1}{a}^i{X}^{n-i-1}}$. From this we calculate the derivative number:   ${\displaystyle (X^n{)}^{\prime }(a)=\underset{x\to a}{\lim }{m}_a(x)=\sum _{i=0}^{n-1}{a}^i{X}^{n-i-1}(a)=\sum _{i=0}^{n-1}{a}^i{a}^{n-i-1}=n{a}^{n-1}}$.   The reciprocal function  ${\displaystyle \frac{1}{X}}$ is differentiable at every $a\ne 0$ with   ►   ${\displaystyle (\frac{1}{X}{)}^{\prime }(a)=-\frac{1}{a^2}}$ [7.3.4] Proof: According [7.2.4] ${\displaystyle -\frac{1}{aX}}$ is the continuous extension of  ${\displaystyle m_a}$ at a. So we get the following derivative number   ${\displaystyle (\frac{1}{X}{)}^{\prime }(a)=\underset{x\to a}{\lim }{m}_a(x)=-\frac{1}{aX}(a)=-\frac{1}{a^2}}$ .   The root function  $\sqrt{X}$ is differentiable at every $a>0$ and   ►   ${\displaystyle {\sqrt{X}}^{\prime }(a)=\frac{1}{2\sqrt{a}}}$ [7.3.5] Proof: [7.2.5] shows that  ${\displaystyle m_a}$ is continuously extended at a by ${\displaystyle \frac{1}{\sqrt{X}+\sqrt{a}}}$, yielding   ${\displaystyle {\sqrt{X}}^{\prime }(a)=\underset{x\to a}{\lim }{m}_a(x)=\frac{1}{\sqrt{X}+\sqrt{a}}(a)=\frac{1}{2\sqrt{a}}}$ as derivative number.   The root function  $\sqrt{X}$ is not differentiable at 0. The difference quotient function ${\displaystyle m_0=\frac{1}{\sqrt{X}}}$ is not continuously extendable at 0. We see this from ${\displaystyle (\frac{1}{n^2})}$, a sequence in ${ℝ}^{>0}$ converging to 0, with the image sequence ${\displaystyle (m_0(\frac{1}{n^2}))=(n)}$ being divergent.   The absolute value function  $|X|$  is not differentiable at 0:  ${\displaystyle \frac{|X|}{X}}$ (cf. [7.2.6]) can't be continuously extended. Take e.g. the zero sequence ${\displaystyle (\frac{{(-1)}^n}{n})}$ in ${ℝ}^{\ne 0}$. The image sequence $({(-1)}^n)$ is divergent.

Consider:

• According [7.3.2] especially all the constant functions c and the identity function X as well are differentiable everywhere with the following derivative numbers:
   

  ${\displaystyle \begin{array}{c}{c}^{\prime }(a)=0\\ X^{\prime }(a)=1\text{.}\end{array}}$

  [7.3.6]

  
   

We started our issue looking for tangents. Having set up the derivative numbers the main problem, getting the correct gradient, is now solved. So we are prepared to build tangents.

Definition:   Let   $f:A\to ℝ$ be differentiable at $a\in A$. We call the function   ${\displaystyle t_a≔f(a)+(X-a)f^{\prime }(a)}$ [7.3.7]   the tangent function of  f with respect to a.

Consider:

• Occasionally we write  ${\displaystyle t_{f,a}}$ instead of  ${\displaystyle t_a}$ to point to the relation to  f.

• ${\displaystyle t_a}$ is a linear function, touching the graph of  f in (a, f(a)). Note that ${\displaystyle t_a(a)=f(a)}$, and that $f^{\prime }(a)$ is the gradient factor of  ${\displaystyle t_a}$.

• Independently from A tangent functions always have the whole of $ℝ$ as their domain.

Working with tangents often comprises the search for the so called normal of  f, i.e. the perpendicular to the tangent running through (a, f(a)). There is a linear function to represent non-vertical normals:

Consider:

• If necessary we use the more detailed notation ${\displaystyle n_{f,a}}$ for the normal function.

• As ${\displaystyle n_a(a)=f(a)}$  ${\displaystyle n_a}$  and f actually intersect at a. Its slope is the reciprocal of the slope of ${\displaystyle t_a}$ with a reversed sign. ${\displaystyle n_a}$ is thus perpendicular

  i

  Rotating a straight line g with non-zero slope ${\displaystyle m=\frac{h}{l}\ne 0}$ by 90° results in a perpendicular line ${\displaystyle g^{\perp }}$ with a slope of ${\displaystyle \frac{l}{-h}=-\frac{1}{m}}$

  to ${\displaystyle t_a}$.

• Independently from A normal functions always have the whole of ${\displaystyle ℝ}$ as their domain.

As an example we calculate the tangent and normal function for the cubic function ${\displaystyle {\mathrm{X}}^3}$ with respect to 1. With ${\displaystyle {\mathrm{X}}^3(1)=1}$ and ${\displaystyle ({\mathrm{X}}^3{)}^{\prime }(1)=3\cdot 1^2=3}$ we have ${\displaystyle \begin{array}{l}{t}_1=1+(\mathrm{X}-1)\cdot 3=3\mathrm{X}-2\\ n_1=1-(\mathrm{X}-1)\cdot \frac{1}{3}=-\frac{1}{3}\mathrm{X}+\frac{4}{3}\end{array}}$

It is an advantage to switch to the vector notation and represent a tangent as a line. We only need to know one point of that line, (a, f(a)) is the most obvious one, and to calculate a directional vector. Having in mind that for an horizontal increment of 1 the derivative ${\displaystyle f^{\prime }(a)}$ is the appropriate vertical increment we find our vector representation as follows:

As both vectors  ${\displaystyle \mathrm{\left(}\begin{array}{c}{f}^{\prime }(a)\\ -1\end{array}\mathrm{\right)}}$  and  ${\displaystyle \mathrm{\left(}\begin{array}{c}1\\ f^{\prime }(a)\end{array}\mathrm{\right)}}$  are perpendicular to each other the normal could be represented as

[7.3.10] has the additional advantage that vertical normals are included as well. Consider however that in the non-vertical case ${\displaystyle \mathrm{\left(}\begin{array}{c}{f}^{\prime }(a)\\ -1\end{array}\mathrm{\right)}}$  and  ${\displaystyle \mathrm{\left(}\begin{array}{c}1\\ -\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$f^{\prime }(a)$}\right.\end{array}\mathrm{\right)}}$ represent the same direction.

For an example we return to ${\displaystyle {\mathrm{X}}^3}$. The tangent of ${\displaystyle {\mathrm{X}}^3}$ with respect to 1 is the line ${\displaystyle \mathrm{\left(}\begin{array}{c}1\\ 1\end{array}\right)+<\left(\begin{array}{c}1\\ 3\end{array}\mathrm{\right)}>}$ and its normal is ${\displaystyle \mathrm{\left(}\begin{array}{c}1\\ 1\end{array}\right)+<\left(\begin{array}{c}3\\ -1\end{array}\mathrm{\right)}>}$.

7.2.

7.4.