3.1 Ordinary Derivatives

In "ordinary" calculus, one deals with functions of a single independent variable. This is usually illustrated by an expression such as:

$$y = f(x)$$ (3.1.1)

where x is the independent variable and y is the dependent variable.

Another way to look at this is to imagine that f(x) is a machine. We drop in a value of x, the machine churns for a while and out pops the result y. What goes on inside the machine depends on the actual function used. The notation f(x) is a formal notation. That is, you can't really do anything with it (except look at it). To use the machine you must supply a concrete function.

Here's an example:

$$y = 4x^2 + 3x - 7$$ (3.1.2)

Insert any value for x in that formula and (after some work) a value of y will pop out. For instance, insert 2 and 15 pops out as the result.^3.1 Of course, some values of the independent variable might break the machine. An example is

$$y = f(x) = \ln (x)$$ (3.1.3)

where a negative value for x will cause the machine to complain loudly. The range of allowed values for the independent variable is often called just that, the range.

Now we could ask "if I change x by a little bit, by how much does y change?"^3.2 The answer to that question is the derivative, usually denoted by

$$z = \frac{dy}{dx}$$

where z is the answer to that question.