In the seventeenth century, mathematicians and other thinkers were confronted with new problems for which they had no solution methods:
(a) The problem of instantaneous speed and acceleration.
(b) The problem of instantaneous variation in the direction of motion.
(c) The problem of the maximum and minimum values of a function.
(d) The problem of determining nonlinear lengths, areas under a curve and volumes of revolution.
These problems and many others were hard nuts to crack for the mathematicians and other thinkers of the seventeenth century. The contributions made by Newton and Leibniz in the search for solution methods for these types of problems heralded the belief that a basic concept underlies these and other similar problems. This concept is the instantaneous rate of change of one variable with respect to another. The branch of mathematics that utilizes this concept is known as differential calculus.
There are two basic concepts inherent in differential calculus: change and rate of change. The concept of change is all around us: we change our clothes and diet frequently; our bodies change over time; the water level of a boiling water in a pot changes over time; the height above the ground of a ball thrown up into the air changes; etc. However, the mere fact of change is not sufficient for the analysis of relational phenomena which are often expressed as functions. It becomes necessary in such scenarios to determine the rate of change of one variable with respect to another. For example, in the case of the ball thrown into the air, one may be interested in the rate of change of height with respect to time. Many important human measurements are measurements of rate of change: annual revolution of the earth around the sun; metabolic rate which is the rate of oxygen consumption per second; the number of planes in the sky per unit time; the number of calls arriving at a phone tower per unit time; flow rates etc.
The rate of change is broadly classified into two types in differential calculus: average rate of change and instantaneous rate of change. In many instances, estimation of the average rate of change is sufficient. For example, the speed measured by the odometer is an average rate of change of distance with respect to time. In many scientific phenomena, the instantaneous rate of change is the rate of change of interest. Mathematically, an instant is a given value of t as opposed to an interval which is some range of t-values. The mathematical tool used to measure instantaneous rate of change is called the derivative and the process of computing the derivative is called differentiation.
The equation in figure 1.1 is the derivative of y with respect to x. This equation simply states that the instantaneous rate of change of y with respect to x at x is equal to f'(x). The two common mathematical representations of this derivative are: f'(x) and dy/dx. The variable x is the independent variable of the function f and the variable y is the dependent variable. In many applications, t is the independent variable and the instantaneous rate of change is with respect to time. There are many differentiation rules for computing derivatives.
All problems of differential calculus are concerned with rate of change. Consequently, differential calculus is an expression of Pj Problems.