Higher Concepts of Mathematics CALCULUScurve y = f(x) at point P.The tangent line has the slope of the curve dy/dx, where, q is the angle between the tangent lineAB and a line parallel to the x-axis. But, tan q also equals Dy/Dx for the tangent line AB, andDy/Dx is the slope of the line. Thus, the slope of a curve at any point equals the slope of the linedrawn tangent to the curve at that point. This slope, in turn, equals the derivative of y withrespect to x, dy/dx, evaluated at the same point.These applications suggest that a derivative can be visualized as the slope of a graphical plot.A derivative represents the rate of change of one quantity with respect to another. When therelationship between these two quantities is presented in graphical form, this rate of changeequals the slope of the resulting plot.The mathematics of dynamic systems involves many different operations with the derivatives offunctions. In practice, derivatives of functions are not determined by plotting the functions andfinding the slopes of tangent lines. Although this approach could be used, techniques have beendeveloped that permit derivatives of functions to be determined directly based on the form of thefunctions. For example, the derivative of the function f(x) = c, where c is a constant, is zero.The graph of a constant function is a horizontal line, and the slope of a horizontal line is zero.f(x) = c(5-6)d[f(x)]dx0The derivative of the function f(x) = ax + c (compare to slope m from general form of linearequation, y = mx + b), where a and c are constants, is a. The graph of such a function is astraight line having a slope equal to a.f(x) = ax + c(5-7)d[f(x)]dxaThe derivative of the function f(x) = axn, where a and n are constants, is naxn-1.f(x) = axn(5-8)d[f(x)]dxnaxn 1The derivative of the function f(x) = aebx, where a and b are constants and e is the base of naturallogarithms, is abebx.Rev. 0 Page 37 MA-05
Integrated Publishing, Inc. - A (SDVOSB) Service Disabled Veteran Owned Small Business
