The first derivative of a function at the point x is the slope of the tangent line at x. All linear functions have a constant derivative because the tangent at every point is the line itself. For instance, the linear function fx = 3x + 2 shown in Figure I.1.1 has first derivative 3. But non-linear functions have derivatives whose value depends on the point x at which it is measured. For instance, the quadratic function fx = 2×2 + 4x + 1 has a first derivative that is increasing with x.
It has value 0 at the point x = −1, a positive value when x > −1, and a negative value when x < −1. This section defines the derivatives of a function and states the basic rules that we use to differentiate functions. We then use the first and second derivatives to define properties that are shared by many of the functions that we use later in this book, i.e. the monotonic and convexity properties.
That is, we take the slope of the chord between two points a distance x apart and see what happens as the two points get closer and closer together. When they touch, so the distance between them becomes zero, the slope of the chord becomes the slope of the tangent, i.e. the derivative. This is illustrated in Figure I.1.7. The graph of the function is shown by the black curve.
The chord from P to Q is the dark grey line. By definition of the slope of a line (i.e. the vertical height divided by the horizontal distance), the slope of the chord is: fx + x − fx x The tangenx, by definition. Now we let the increment in x, i.e. x, get smaller the slope of the chord gets closer to the slope of the tangent, i.e.
the derivative. In the limit, when x=0, the points P and Q coincide and the slope of the chord, i.e. the right-hand side of (I.1.14), is equal to the slope of the tangent, i.e. the left-hand side of (I.1.14).