Mathematical Representation:

When the graph of a function is not a straight line then the slope of the plot is different on different points.  The slope of a curve at any point is described as the slope of a line drawn tangent to the curve at that particular point. Above figure shows a line drawn tangent to a curve. A tangent line is basically a line which touches the curve at just one point.  The line AB is tangent to the curve y = f(x) at point P.

The tangent line has the slope of the curve dy/dx, where, Θ is the angle among the tangent line AB and a line parallel to the x-axis.  But, tan Θ also equals Δy/ Δx for the tangent line AB, and Δy/ Δx is the slope of the line.  Therefore, the slope of a curve at any point equals the slope of the line drawn tangent to the curve at that point.   That slope, in turn, equals the derivative of y along with respect to x, dy/dx, evaluated at the similar point.

These applications suggest in which a derivative can be visualized as the slope of a graphical plot. Below derivative represents the rate of change of one quantity with respect to another.   While the relationship among these two quantities is presented in graphical form, that rate of change equals the slope of the resulting plot.

The mathematics of dynamic systems includes several different operations with the derivatives of functions.  By practice, derivatives of functions are not determined through plotting the functions and searching the slopes of tangent lines.  While this approach could be used, techniques have been developed that allow 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. A graph of a constant function is a horizontal line, and the slope of a horizontal line is zero.

f(x) = c

d[f(x)]/dx = 0

The derivative of the function f(x) = ax + c (as compare to slope m from common 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 that is equal to a.

f(x) = ax + c

d[f(x)]/dx =a

The derivative of the function f(x) = axⁿ, where a and n are constants, is nax^n-1.

f(x) = axⁿ

d[f(x)]/dx = nax^n-1

The derivative of the function f(x) = ae^bx, where a and b are constants and e is the base of natural logarithms, is abe^bx.

f (x ) = ae^bx

d[f(x)]/dx = abe^bx

These common techniques for searching the derivatives of functions are significant for those who perform detailed mathematical calculations for dynamic systems. For instance, the designers of nuclear facility systems require an understanding of these techniques, since these techniques are not encountered in the day-to-day operation of a nuclear facility. As an output, the operators of these facilities should know what derivatives are in terms of a rate of change and a slope of a graph, but they will   not normally be needed to search the derivatives of functions.