Differentiability

If the function [math]\displaystyle{ f: D \to {\mathbb R} }[/math] is differentiable in [math]\displaystyle{ x_0\in D }[/math] then there is a function [math]\displaystyle{ f_1: D \to {\mathbb R} }[/math] that is continuous in [math]\displaystyle{ x_0 }[/math] such that

[math]\displaystyle{ f(x) = f(x_0) + (x-x_0) f_1(x) \,. }[/math]

Drag the point [math]\displaystyle{ x }[/math] to draw the function [math]\displaystyle{ f_1 }[/math].

The underlying JavaScript code

board = JXG.JSXGraph.initBoard('box', {
    boundingbox: [-5, 10, 7, -6], 
    axis: true,
    showClearTrace: true,
    showFullscreen: true});

var p = [];
p[0] = board.create('point', [-1,0], {withLabel: false, size:2, color:'blue'});
p[1] = board.create('point', [-0.5,3], {withLabel: false, size:2, color:'blue'});
p[2] = board.create('point', [2,0.5], {withLabel: false, size:2, color:'blue'});
p[3] = board.create('point', [6, 3], {withLabel: false, size:2, color:'blue'});
var pol = JXG.Math.Numerics.lagrangePolynomial(p);
var graph = board.create('functiongraph', [pol, -10, 10], {strokeWidth: 2, name:"f", withLabel: true});

var x0 = board.create('glider', [1, 0, board.defaultAxes.x], {name: 'x_0', size:4});
var fx0 = board.create('point', [
                function() { return x0.X(); }, 
                function() { return pol(x0.X()); }
          ], {name: '', color: 'grey', fixed: true, size:3});
var x = board.create('glider', [5, 0, board.defaultAxes.x], {name: 'x', size:4});
var fx = board.create('point', [
                function() { return x.X(); }, 
                function() { return pol(x.X()); }
          ], {name: '', color: 'grey', fixed: true, size:3});
var line = board.create('line',[fx0, fx],{strokeColor:'#ff0000',dash:2});

var f1 = board.create('point', [
        function() { return x.X(); },
        function() { return (fx.Y()-fx0.Y())/(fx.X()-fx0.X() + 0.0000001); }],
        { size: 1, name: 'f_1', color: 'black', fixed: true, trace: true});
    
var txt = board.create('text', [0.5, 7, function() { 
        return '( ' + 
               fx.Y().toFixed(2) + ' - (' + fx0.Y().toFixed(2) + 
               ') ) / ( ' + 
               fx.X().toFixed(2) + ' - (' + fx0.X().toFixed(2) +
               ') ) = ' + ((fx.Y()-fx0.Y())/(fx.X()-fx0.X())).toFixed(3);
    }]);

board.create('functiongraph',[JXG.Math.Numerics.D(pol)], {dash: 2, name:"f'", withLabel: true});