数学学習

関数の微分可能性

関数が微分可能な場合とその性質を理解する

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

L

Right Derivative: f'+(a)

0.0000

R

✓ 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:

  1. f'-(a) exists (left derivative)
  2. f'+(a) exists (right derivative)
  3. f'-(a) = f'+(a)

If differentiable, then f'(a) = f'-(a) = f'+(a)

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関数の微分可能性 | Maths Learning