Diferenciabilidad de Funciones
Comprender cuándo las funciones son diferenciables y sus propiedades
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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