Derived functions and derivative notation
Given the function x(t) that describes some particular motion, you could plot the corresponding position-time graph, measure its gradient at a variety of times to find the instantaneous velocity at those times and then plot the velocity-time graph. If you had some time left, you might go on to measure the gradient of the velocity-time graph at various times, and then plot the acceleration-time graph for the motion. This would effectively complete the description of the motion, but it would be enormously time consuming and, given the difficulty of reading graphs, not particularly accurate.
Fortunately this graphical procedure can usually be entirely avoided. Starting again from the function x(t), there exists a mathematical procedure, called differentiation, that makes it possible to determine the velocity v_{x}(t) directly, by algebra alone. We shall not try to describe the principles that underpin differentiation, but we will introduce the notation of the subject and list some of the basic results. To make this introduction as general as possible we shall initially consider a general function f(y) rather than the position function x(t).
The central idea is this:
Remember, the gradient of a graph at a given point is defined by the gradient of its tangent at that point.
Given a function f(y) it is often possible to determine a related function of y, called the derived function, with the property that, at each value of y, the derived function is equal to the gradient of the graph of f against y at that same value of y.
The derived function is usually referred to as the derivative of f with respect to y (often abbreviated to derivative) and may be represented by the symbol or, more formally . The notation is reminiscent of the notation that was used when discussing the gradient of a straight line and thus provides a clear reminder of the link between the derived function and the gradient of the f against y graph. However, it is important to remember that is a single symbol representing the derived function, it is not the ratio of two quantities df and dy.
Although there are systematic ways of finding derived functions from first principles, you will not be required to use them in this course. Indeed, physicists are rarely required to do this because tables of derivatives already exist for all the well-known functions, and derivatives of more complicated functions can usually be expressed as combinations of those basic derivatives. Table 6 lists a few of the basic derivatives along with the simplest of the rules for combining them - it also gives some explicit examples of functions and their derivatives. Computer packages are now available that implement the rules of differentiation, these are often used to determine the derivatives of more complicated functions (Figure 32).
Table 6: Some simple derivatives. The functions f, g and h depend on the variable y. The quantities A and n are constants, which may be positive, negative or zero. Note that n is not necessarily an integer
Function f(y) | Derivative | Example |
---|---|---|
The idea of a derivative may be new to you and, if so, may seem rather strange. However, if you know the explicit form of a function, then there are several crucial advantages in using derivatives to determine gradients, rather than making measurements on a graph. In particular, looking up the derivative of a function in a table should be completely accurate, whereas measuring the gradient of the tangent to a graph is always approximate. For example, if f(y) = y^{2} then the derivative of f(y) is df/dy = 2y and evaluating the derivative at y = 3 to find the gradient at that particular value of y gives 6. This is an exact result that could not have been obtained with such precision from measurements on a graph. Moreover, if we want to know the gradient at many different values of y, all we need to do is to substitute each of those values into the general expression for the derivative, df/dy = 2y. This is much simpler than drawing many different tangents and measuring their individual gradients.