Differentieerbaarheid van Functies
Begrijpen wanneer functies differentieerbaar zijn en hun eigenschappen
Interactive Differentiability Visualization
Welcome to the Differentiability Explorer
This interactive visualization helps you understand when functions are differentiable and why some are not. A function is differentiable at a point if it has a well-defined derivative there, which requires the left and right derivatives to be equal.
What you can explore:
- Smooth functions - Differentiable everywhere (f(x) = x²)
- Corner points - Not differentiable where left and right derivatives differ (f(x) = |x|)
- Cusps - Vertical tangent where derivative approaches infinity (f(x) = x^(2/3))
- Vertical tangents - Infinite derivative (f(x) = x^(1/3))
- Discontinuities - Not differentiable if not continuous (step function)
- Absolute value functions - Corner points where differentiability fails
How to Use This Visualization
Interactive Features:
- • Select Function Type - Choose from 6 different functions to explore various differentiability scenarios
- • Adjust Point a - Use the slider to test differentiability at different points
- • Toggle Visualizations - Show/hide left/right tangents, approach points, and derivative information
- • Compare Derivatives - See left and right derivatives side by side
What You'll See:
- • Function Curve (colored) - The original function
- • Point a (red) - The point where differentiability is being tested
- • Left Tangent (blue, dashed) - Tangent line from the left
- • Right Tangent (amber, dashed) - Tangent line from the right
- • Single Tangent (green) - When left and right derivatives are equal
- • Left/Right Approach Points (blue/amber) - Points approaching from each side
- • Differentiability Status - Real-time check with explanation
Smooth curve - differentiable everywhere
Adjust to test differentiability at different points
Differentiability Analysis at x = 0.00
Left Derivative: f'-(a)
0.0000
Right Derivative: f'+(a)
0.0000
✓ Function is DIFFERENTIABLE at x = 0.00
Left and right derivatives are equal
Function Information
Point a: 0.00
f(a): 0.0000
Differentiable: Yes
Key Concepts
Differentiability: A function is differentiable at a point if the left and right derivatives exist and are equal.
Not Differentiable: Functions fail to be differentiable at corners, cusps, vertical tangents, or discontinuities.
Continuity Requirement: If a function is differentiable at a point, it must be continuous there. However, continuity does not guarantee differentiability.
Definition of Differentiability
A function f is differentiable at x = a if and only if:
- f'-(a) exists (left derivative)
- f'+(a) exists (right derivative)
- f'-(a) = f'+(a)
If differentiable, then f'(a) = f'-(a) = f'+(a)
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