Differenzierbarkeit von Funktionen

Verstehen, wann Funktionen differenzierbar sind und ihrer Eigenschaften

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

Function Type

Smooth curve - differentiable everywhere

Point a: 0.00

Adjust to test differentiability at different points

Show Left/Right ApproachShow Tangent LinesShow Derivative Information

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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