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 $d\left(sinx\right)\u2215dx=cosx$ 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 ($\Delta y$) and abscissae ($Deltax)$ between the movable red and the fixed blue point. The magenta coloured point indicates the value of the difference quotient $\Delta y\u2215\Delta x$. 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 $\Delta x\to 0$ 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.

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:

$$\frac{d}{dx}\left(f\left(x\right)+c\right))=\frac{d}{dx}f\left(x\right)$$

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

$$\frac{{d}^{2}}{d{x}^{2}}\left(f\left(x\right)+{c}_{1}+{c}_{2}x\right)=\frac{{d}^{2}}{d{x}^{2}}f\left(x\right).$$

The second derivative (the curvature of the original function) is identical for all functions, that only differ by a constant ${c}_{1}$ and a linear term ${c}_{2}x$.

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 ${c}_{2}x$ to the sine function. This results in the first derivative being shifted by the value ${c}_{2}$ 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 ${c}_{2}$.

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.