This Quantum World/Appendix/Calculus

Differential calculus: a very brief introductionEdit

Another method by which we can obtain a well-defined, finite number from infinitesimal quantities is to divide one such quantity by another.

We shall assume throughout that we are dealing with well-behaved functions, which means that you can plot the graph of such a function without lifting up your pencil, and you can do the same with each of the function's derivatives. So what is a function, and what is the derivative of a function?

A function   is a machine with an input and an output. Insert a number   and out pops the number   Rather confusingly, we sometimes think of   not as a machine that churns out numbers but as the number churned out when   is inserted.


 


The (first) derivative   of   is a function that tells us how much   increases as   increases (starting from a given value of   say  ) in the limit in which both the increase   in   and the corresponding increase   in   (which of course may be negative) tend toward 0:

 

The above diagrams illustrate this limit. The ratio   is the slope of the straight line through the black circles (that is, the   of the angle between the positive   axis and the straight line, measured counterclockwise from the positive   axis). As   decreases, the black circle at   slides along the graph of   towards the black circle at   and the slope of the straight line through the circles increases. In the limit   the straight line becomes a tangent on the graph of   touching it at   The slope of the tangent on   at   is what we mean by the slope of   at  

So the first derivative   of   is the function that equals the slope of   for every   To differentiate a function   is to obtain its first derivative   By differentiating   we obtain the second derivative   of   by differentiating   we obtain the third derivative   and so on.

It is readily shown that if   is a number and   and   are functions of   then

   and   

A slightly more difficult problem is to differentiate the product   of two functions of   Think of   and   as the vertical and horizontal sides of a rectangle of area   As   increases by   the product   increases by the sum of the areas of the three white rectangles in this diagram:


 


In other "words",

 

and thus

 

If we now take the limit in which   and, hence,   and   tend toward 0, the first two terms on the right-hand side tend toward   What about the third term? Because it is the product of an expression (either   or  ) that tends toward 0 and an expression (either   or  ) that tends toward a finite number, it tends toward 0. The bottom line:

 

This is readily generalized to products of   functions. Here is a special case:

 

Observe that there are   equal terms between the two equal signs. If the function   returns whatever you insert, this boils down to

 

Now suppose that   is a function of   and   is a function of   An increase in   by   causes an increase in   by   and this in turn causes an increase in   by   Thus   In the limit   the   becomes a   :

 


We obtained   for integers   Obviously it also holds for   and  

  1. Show that it also holds for negative integers   Hint: Use the product rule to calculate  
  2. Show that   Hint: Use the product rule to calculate  
  3. Show that   also holds for   where   is a natural number.
  4. Show that this equation also holds if   is a rational number. Use  

Since every real number is the limit of a sequence of rational numbers, we may now confidently proceed on the assumption that   holds for all real numbers