The simulation demonstrates the limit process in deriving the first derivative of a sine function with a superimposed linear term (a straight line) . For this case the second derivative should be independent of the linear term.
y = sin(x) +c x
dy/dt(x1) = x2➙x1 lim (y2 - y1 )/(x2 - x1) = cos(x1)+c
d 2y/dt2(x1) = -sin(x1)
A slider defines the blue model point x1 , y1 . Drawing the red point along the function defines the second point x2, y2 in the limit process. The parameter (gradient) c of the linear term is defined by drawing the magenta colored rectangle.
The secant between the 2 points is drawn in black, and is prolongated beyond the second point in green.
The ordinate difference (y2 - y1) is shown as a red arrow, the abscissa difference
(x2 -x1) as a blue one.
The difference quotient (y2 - y1)/(x2 -x1), which corresponds to the tangent of the angle between both components, is shown as a magenta colored point.
The analytic first derivative cos(x) +c = d(sin(x)+cx)/dx is drawn as a beige line, the second derivative sin(x) =d(cos(x)+c)/dx. as a blue one.
Choose an arbitrary model point with the slider. Draw the blue point towards the model point. With this limit process the secant will become the tangent in the model point, the difference quotient will become a point on the beige line ~ a differential quotient.
Change c by drawing the magenta rectangle. The analytic first derivative will be shifted parallel in the y- direction, while the analytic second derivative will stay unaffected.