How to Spot Differentiability from a Graph: A Visual Guide for JEE
Learn to instantly identify differentiable and non-differentiable points from function graphs - essential skill for JEE calculus problems.
Why Graphical Differentiability Matters in JEE
JEE frequently tests differentiability through graph interpretation. Mastering visual identification gives you:
- Quick problem solving without complex calculations
- Better intuition for function behavior
- Time savings of 2-3 minutes per question
- Higher accuracy in MCQs and integer type questions
Smooth & Continuous Curves
A function is differentiable at points where its graph is smooth and has a unique tangent.
Visual Identification:
[Graph: Smooth continuous curve]
No sharp corners, breaks, or vertical tangents
✓ Smooth transition between points
✓ No abrupt changes in direction
✓ Continuous with well-defined slope
Sharp Corners (Cusps)
Functions with abrupt changes in direction create corners where derivatives don't exist.
Visual Identification:
[Graph: Absolute value function |x| at x=0]
Sharp V-shape with point at origin
Example: $f(x) = |x|$ at $x = 0$
Left derivative: $-1$
Right derivative: $+1$
Result: Not differentiable at corner point
Vertical Tangents
When the tangent line becomes vertical, the derivative is undefined (infinite slope).
Visual Identification:
[Graph: Cube root function ∛x at x=0]
Steep curve approaching vertical line
Example: $f(x) = x^{1/3}$ at $x = 0$
Slope approaches: $\pm\infty$
Result: Not differentiable at vertical tangent
Discontinuities & Breaks
A function must be continuous to be differentiable. Breaks automatically mean non-differentiability.
Visual Identification:
[Graph: Piecewise function with jump discontinuity]
Sudden jump in y-values at specific x
Types of discontinuities:
• Jump discontinuities
• Infinite discontinuities
• Removable discontinuities
Rule: Continuous → Differentiable, but not vice versa
🔍 4-Step Visual Analysis Framework
Check Continuity
Look for breaks, jumps, or holes in the graph
Look for Corners
Identify sharp changes in direction (V-shapes)
Check Vertical Tangents
Look for points where curve becomes vertical
Verify Smoothness
Ensure gradual, continuous curvature
Patterns 4-6 Available in Full Version
Includes Oscillatory Behavior, Piecewise Functions, and Parametric Curves with interactive graphs
📝 Quick Visual Test
Identify differentiable/non-differentiable points in these common JEE graphs:
[Graph: f(x) = |x-1| + |x+1|]
At what points is this function NOT differentiable?
[Graph: f(x) = x^(2/3)]
Is this function differentiable at x=0? Why?
[Graph: Piecewise function with smooth connection]
At the connection point, is the function differentiable?
Actual JEE Problem
Let $f(x) = \min\\{|x|, |x-1|, |x+1|\\}$. The number of points where $f(x)$ is not differentiable is:
Visual Analysis Approach:
[Graph: Three V-shaped curves with minimum envelope]
Identify corners in the minimum envelope function
Step 1: Sketch $|x|$, $|x-1|$, $|x+1|$
Step 2: Identify the minimum envelope
Step 3: Count sharp corners in envelope
Answer: 4 points of non-differentiability
Ready to Master Graphical Differentiability?
Get complete access to all 6 patterns with interactive graphs and 25+ practice problems