In the mathematical modelling of real physical problems it is very often a process that we wish to model. That is, one physical quantity changing in response to another.
One example of a process is the temperature within the earth changing as you go deeper. Another is the decay of radioactive nuclei as time passes. With a moment's thought you can think of many many more examples.
To describe mathematically this idea of one quantity depending on another, we use functions. One quantity, say the depth within the earth, is the input, (or independent variable) and the other quantity, say the temperature, is the output (or dependent variable). This should all be familiar to you.
We can now go one step further in our consideration of processes and try to describe the rate at which these processes occur. So, returning to the earth-temperature example, knowing that the temperature increases as we go deeper, we might ask whether the temperature increases at a constant rate the deeper you go, or does it get hotter more and more rapidly?
Similarly for the radioactive decay example, does the decay occur more and more rapidly, or does it slow down as the number of remaining nuclei falls, or does it always occur at a constant rate? (In this particular case, the rate of decay is actually proportional to the number of nuclei that are left, so the decay gets slower and slower. This is exponential behaviour.)
To study the rate at which processes occur, we use
"differentiation". This means that we start with the function
that represents the process, then by carrying out a standard procedure on
it we find the rate at which that particular function changes. So that
gives us the rate at which the process occurs.
The "standard
procedure" that we carry out is called differentiating the function,
and the resulting rate at which the function changes (which is itself
another function) is called the derivative.
Why does the rate of change itself have to be another function?
In this Notebook we'll look at what this procedure of differentiation
involves and why, and some commonly-used results that follow from it. In
doing this we'll differentiate all the functions we know and so get to
understand them better.
By the way, in case you're wondering why
it's called "differentiation", it's because we get the result by
thinking about small differences, as you'll see in the next few
sections.