Graphical Interpretations of Derivative

To animate a series of graphs, double click on any one of the graphs. You can control the speed of the animation using the keypad or the controls at the bottom left of the window.

One Definition

Here is another of the definitions of derivative.

\(\(Lim\+\(\(h\)\(\ \)\(->\)\(\ \)\(0\)\(\ \)\)\) \(f \((a + h)\) - f \((a)\)\)\/h\)

Here is some Mathematica code to animate the graphical interpretation of this equation as h goes to zero.

f[x_]  =  x^3 -  x^2/2 - 15  x  + 2;

    
a  =  2;
y[x_, 
          h_]  =  (f[a + h] - f[a])/(h) *(x - (a))  +  
          f[a];
ytangent[x_]  =  f'[a] (x - a) + 
          f[a];

    
Table[
      Plot[{f[x], y[x, h], ytangent[x]}, {x, -8, 8}, 
        PlotRange -> {{0, 8}, {-30, 30}}, 
        PlotStyle -> {RGBColor[0, 1, 0], RGBColor[1, 0, 0], 
            RGBColor[0, 0, 1]}], {h, 2, 0.1, -0.1}]

Second Definition

Here is one of the definitions of derivative.

\(\(Lim\+\(\(h\)\(\ \)\(->\)\(\ \)\(0\)\(\ \)\)\) \(f \((a + h)\) - f \((a - h)\)\)\/\(2  h\)\)

Here is some Mathematica code to animate the graphical interpretation of this equation as h goes to zero.

f[x_]  =  x^3 -  x^2/2 - 15  x  + 2;

    
a  =  2;
y[x_, 
          h_]  =  (f[a + h] - f[a - h])/(2  
                  h) *(x - (a + h))  +  
          f[a + h];
ytangent[x_]  =  f'[a] (x - a) + 
          f[a];

    
Table[
      Plot[{f[x], y[x, h], ytangent[x]}, {x, -8, 8}, 
        PlotRange -> {{0, 8}, {-30, 30}}, 
        PlotStyle -> {RGBColor[0, 1, 0], RGBColor[1, 0, 0], 
            RGBColor[0, 0, 1]}], {h, 2, 0.1, -0.1}]