Higher Concepts of Mathematics
curve 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 line
AB and a line parallel to the x-axis. But, tan q also equals Dy/Dx for the tangent line AB, and
Dy/Dx is the slope of the line. Thus, the slope of a curve at any point equals the slope of the line
drawn tangent to the curve at that point. This slope, in turn, equals the derivative of y with
respect 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 the
relationship between these two quantities is presented in graphical form, this rate of change
equals the slope of the resulting plot.
The mathematics of dynamic systems involves many different operations with the derivatives of
functions. In practice, derivatives of functions are not determined by plotting the functions and
finding the slopes of tangent lines. Although this approach could be used, techniques have been
developed that permit derivatives of functions to be determined directly based on the form of the
functions. 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
The derivative of the function f(x) = ax + c (compare to slope m from general form of linear
equation, y = mx + b), where a and c are constants, is a. The graph of such a function is a
straight line having a slope equal to a.
f(x) = ax + c
The derivative of the function f(x) = axn, where a and n are constants, is naxn-1.
f(x) = axn
The derivative of the function f(x) = aebx, where a and b are constants and e is the base of natural
logarithms, is abebx.