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Differential Calculus |
Let the dependent variable y be a function of the independent variable x, expressed by y = f(x). If x0 is a value of x in its domain of definition, then y0 = f(x0) is the corresponding value of y. Let h and k be real numbers, and let y0 + k = f(x0 + h). (Δx, read “delta x,” is used quite frequently in place of h.) When Δx is used in place of h,Δy is used in place of k. Then clearly
and
This ratio is called a difference quotient. Its intuitive meaning can be grasped from the geometrical interpretation of the graph of y = f(x). Let A and B be the points (x0, y0), (x0 + h, y0 + k), respectively, as in the Derivatives illustration. Draw the secant AB and the lines AC and CB, parallel to the x and y axes, respectively, so that h = AC, k = CB. Then the difference quotient k/h equals the tangent of angle BAC and is therefore, by definition, the slope of the secant AB. It is evident that if an insect were crawling along the curve from A to B, the abscissa x would always increase along its path but the ordinate y would first increase, slow down, then decrease. Thus, y varies with respect to x at different rates between A and B. If a second insect crawled from A to B along the secant, the ordinate y would vary at a constant rate, equal to the difference quotient k/h, with respect to the abscissa x. As the two insects start and end at the same points, the difference quotient may be regarded as the average rate of change of y = f(x) with respect to x in the interval AC.
If the limit of the ratio k/h exists as h approaches 0, this limit is called the derivative of y with respect to x, evaluated at x = x0. For example, let y = x2 and x = 3, so that y = 9. Then 9 + k = (3 + h)2; k = (3 + h)2 - 9 = 6h + h2; k/h = 6 + h; and
Referring back to the Derivatives illustration, the secant AB pivots around A and approaches a limiting position, the tangent AT, as h approaches 0. The derivative of y with respect to x, at x = x0, may be interpreted as the slope of the tangent AT, and this slope is defined as the slope of the curve y = f(x) at x = x0. Further, the derivative of y with respect to x, at x = x0, may be interpreted as the instantaneous rate of change of y with respect to x at x0.
If the derivative of y with respect to x is found for all values of x (in its domain) for which the derivative is defined, a new function is obtained, the derivative of y with respect to x. If y = f(x), the new function is written as y’ or f’(x), Dxy or Dxf(x), (dy)/(dx) or df(x)/dx. Thus, if y = x2, y + k = (x + h)2; k = (x + h)2 - x2 = 2xh + h2; k/h = 2x + h, whence
Thus, as before, y’ = f’(x) = 6 at x = 3, or f’(3) = 6; also, f’(2) = 4, f’(0) = 0, and f’(-2) = -4.
As the derivative f’(x) of a function f(x) of x is itself a function of x, its derivative with respect to x can be found; it is called the second (order) derivative of y with respect to x, and is designated by any one of the symbols y” or f”(x), Dx2y or Dx2f(x), (d2y)/(dx2) or (d2f(x))/(dx2). Third- and higher-order derivatives are similarly designated.
Every application of differential calculus stems directly or indirectly from one or both of the two interpretations of the derivative as the slope of the tangent to the curve and as the rate of change of the dependent variable with respect to the independent variable. In a detailed study of the subject, rules and methods developed by the limit process are provided for rapid calculation of the derivatives of various functions directly by means of various known formulas. Differentiation is the name given to the process of finding a derivative.
Differential calculus provides a method of finding the slope of the tangent to a curve at a certain point; related rates of change, such as the rate at which the area of a circle increases (in square feet per minute) in terms of the radius (in feet) and the rate at which the radius increases (in feet per minute); velocities (rates of change of distance with respect to time) and accelerations (rates of change of velocities with respect to time, therefore represented as second derivatives of distance with respect to time) of points moving on straight lines or other curves; and absolute and relative maxima and minima.
