Differential calculus involves the analysis of functions, specifically, determining their instantaneous rates of change. An important feature of any function is its rate of change. Geometrically, rate of change is associated with the graph of a function. The rate of change of a straight line (the simplest kind of real valued function) is the slope of the line. The slope is defined as the ratio of the vertical change, or "rise," to the horizontal change, or "run," that occurs between any two points on the line. Because the slope is the same between any two points, the rate of change of such a function is said to be constant. In general, however, any function whose graph is not a straight line has a varying rate of change. The rate of change in the vicinity of a particular point on the graph of curve can be approximated by drawing a straight line through two points in the neighborhood of that point, and determining the slope of the line. Suppose we are interested in the rate of change at the point (x,f(x)). First, choose a second nearby point, say (x+h, f(x+h)). Then the slope of the line segment connecting these two points is [f(x+h) - f(x)] 4 [(x+h) - x]. The shorter the approximating line segment becomes, the more accurate the approximation of the rate of change at the point (x, f(x)) becomes. In the limit that h approaches zero the slope of the approximating line segment becomes exactly the rate of change of the function at the point (x,f(x)). Thus, the instantaneous rate of change of a function, called the derivative of the function, is defined by:
where the notation df(x) is intended to indicate that the derivative is the ratio of an infinitesimal change in f(x) (the rise) to the corresponding infinitesimal change in x (the run). The derivative of a function is itself a function, and so may also have a derivative. Often times the derivative of the derivative is an important quantity. Called the second derivative of the original function f, it is denoted by f"(x).
An important application of differential calculus involves using information about the first and second derivatives, and the appropriate geometric interpretations, to graph functions. For example, the first derivative of a function is its rate of change. The value of the first derivative at a given point is equal to the slope of the tangent to the graph of the function at that point. When the derivative is positive, the function is said to be increasing (the value of the function increases with increasing x). When the derivative is negative, the function is said to be decreasing (the value of the function decreases with increasing x). When the value of the derivative is zero at a point, the tangent is horizontal, and the function changes from increasing to decreasing, or vice versa, depending on the sign of the second derivative. The second derivative is the rate of change of the rate of change, and thus contains information about the curvature of the function. When the second derivative is positive, the function is concave upward (as though it would hold water). When the second derivative is negative, the function is concave downward. With this knowledge, and a few points in the function, a reasonable graph can be drawn without having to plot hundreds of points.
Other applications of differential calculus include the solution of rate problems and optimization problems. In general, a rate is the ratio of change in one quantity to the simultaneous change in a second quantity. Thus, the derivative, being an instantaneous rate, is applicable to any problem in which the rate of change of one quantity with respect to another is of interest. Innumerable applications in engineering and science affect our daily lives. For instance, the instantaneous velocity of an orbiting communications satellite is calculated from knowledge of its position as a function of time. The acceleration of a falling body is calculated from knowledge of its velocity as a function of time, which in turn is calculated from knowledge of its position as a function of time. The force required to deliver natural gas through a pipeline, over large distances, is calculated using the derived of the gas pressure with respect to distance.
Optimization problems are problems that require knowledge of maximum or minimum values of functional relationships. For example, it can be shown that a sphere has the least surface area for a given volume of any geometric solid. Thus, the optimum shape for a raindrop is spherical because this shape contains the most water, but has the least amount of surface area, hence the least surface energy (a measure of the work required to form the drop).