DERIVATIVE

in mathematics, the rate of change, or instantaneous velocity, of a function with respect to a variable. Geometrically, the derivative of a function can be interpreted as the slope of the graph of the function or, more precisely, as the slope of the tangent line at a point. Its calculation, in fact, derives from the slope formula for a straight line, except that a limiting process must be used for curves. The formula for the slope of a straight line is (y1 - y0)/(x1 - x0), or [ f (x0 + h) - f (x0)]/h, if h is used for x1 - x0 and f (x) for y (see Graph 1). For a curve, this ratio will depend upon where the points are chosen, reflecting the idea that a curve has a different slope at different points. To find the slope at a desired point, the choice of the second point needed to calculate the ratio represents a difficulty, because, in general, if the second point is far from the first, the ratio will represent an average slope along the portion of the curve cut off, rather than the slope at either point (see Graph 2). To get around this difficulty, a limiting process is used whereby the second point is not fixed but specified by a variable, as h in the ratio for the straight line above. Finding the limit (q.v.) in this case is a process in which a number is found that the ratio (also called the difference quotient) approaches as h approaches 0, so that the limiting ratio will represent the actual slope at the given point. Some manipulations must be done on the quotient [ f (x0 + h) - f (x0)]/h so that it can be rewritten in a form in which the limit as h approaches 0 can be seen more directly. In finding the derivative of x2 when x is 2, the quotient is [(2 + h)2 - 22]/h. By expanding the numerator, the quotient becomes (4 + 4h + h2 - 4)/h = (4h + h2)/h. Both numerator and denominator still approach 0, but if h is not actually zero but only very close to it, then h can be divided out, giving 4 + h, which is easily seen to approach 4 as h approaches 0. To sum up, the derivative of f (x) at x0, written as f (x0), (df/dx)(x0), or Df (x0), is defined as if this limit exists. Differentiationi.e., calculating the derivativeseldom requires the use of the basic definition but can instead be accomplished through a knowledge of the three basic derivatives, the use of four rules of operation, and a knowledge of how to manipulate functions. See differentiation. Derivatives are involved in all phases of calculus and differential equations and find applications in problems of velocity, maxima, curve analysis, and approximations.