5.6 The Limiting process for Obtaining the Differential Quotient

After these preliminary discussions we want to visualize the limiting process involved in differentiation in a simulation for the sine function. Fig.5.7 shows the sine function y = sinx over somewhat more than a full period. The first (analytical) derivative , the cosine function d(sinx)dx = cosx is drawn in yellow. A blue point, at which the limiting process will be observed, can be adjusted with the slider on the plot of the sine function. The large red point can also be adjusted along the sine curve. The line connecting these two points is extended in green.

The red and blue arrows show the difference of the ordinates (Δy) and abscissae (Deltax) between the movable red and the fixed blue point. The magenta coloured point indicates the value of the difference quotient ΔyΔx. If you pull the red point to the blue point the line connecting them becomes the tangent and the point for the difference quotient moves to the curve for the first derivative. This is the limiting process Δx 0 of the difference quotient. You can reconstruct the curve of the first derivative via moving the blue computation point along the sine curve. In the description pages you will find hints for further useful experiments.

Figure 5.7: Visualization of how the difference quotient approaches the differential quotient in the limit of Δx 0 for the example of the sine function (black) and its first derivative (yellow). The position of the computation point in blue can be changed with the slider and the red point can be moved with the mouse. The small point coloured in magenta indicates the value of the respective difference quotient. With decreasing width of the abscissa interval it approaches the analytical differential quotient.

The difference quotient is obviously does not change, if the the curve that is drawn symmetrically to the magenta coloured x-axis is moved up or down by a constant value c; the same applies to the differential quotient. This corresponds to the rule, that the derivative of a constant vanishes. All functions that are different only by a constant value in y-direction have the same derivative:

d dx(f(x) + c)) = d dxf(x)

Applying the same line of thought to the determination of the second derivative (the figure shown above is also valid if one interprets the black curve as the first derivative and the yellow as the second derivative), it follows

d2 dx2(f(x) + c1 + c2x) = d2 dx2f(x).

The second derivative (the curvature of the original function) is identical for all functions, that only differ by a constant c1 and a linear term c2x.

This is visualized via the simulation in Fig.5.8, where the second derivative - sinx is plotted in addition. A blue rectangle is also present, that can be pulled to add a linear term c2x to the sine function. This results in the first derivative being shifted by the value c2 in y-direction. The magenta coloured point of the difference quotient is again lead to the curve of the first derivative via the limiting process. The second derivative is not affected via changing c2.

Figure 5.8: Limiting process for the calculation of the second derivative (blue) for a sine function (black) that has been supplemented by a linear term. The computation point and the width of the interval can be adjusted via the slider and pulling the red point with the mouse and the linear term can be changed via pulling the rectangular purple marker. The first derivative drawn in yellow is then moved in y-direction.

The second derivative characterizes a function up to two constants, that are initial values of the function, namely the value of the function itself and the first derivative at a point chosen as x = 0 without loss of generality. From the cohort of all functions, that have the same second derivative only the initial values determine a unique function.

This train of thought can also be applied to higher derivatives. The n-th derivative characterizes a cohort of curves with n parameters.