Differential calculus: a very brief introduction
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
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",
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
Show that it also holds for negative integers Hint: Use the product rule to calculate
Show that Hint: Use the product rule to calculate
Show that also holds for where is a natural number.
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