Integration

Abstract

In Chapter 2, we learned how to differentiate functions using either of two commands. Both

$$ \begin{gathered} > diff\left( {f\left( t \right),t} \right); \hfill \\ > D\left( f \right)\left( t \right); \hfill \\ \end{gathered} $$

(7.1)

give the same result, namely the first derivative of the function

$$ f: = t - > f(t)$$

(7.2)

The first form, with diff, acts on the functional expression f (t), whereas the second, with D, gives a new mapping. A derivative gives the rate of change of a function with respect to its argument. Thus, the time derivative of the position x of a particle is its time-rate of change, in other words the velocity, and the derivative of the velocity is the acceleration. Similarly the time-rate of change of the energy is the power, and so on. Graphically, the derivative is a new function whose value is the slope of the original one.