Differentiability
Not every function has a derivative everywhere. A function is differentiable at a point if the derivative exists there. Let's see when derivatives don't exist.
When Derivatives Don't Exist
1. Corners/Cusps - Sharp points (like |x| at x=0)
2. Vertical Tangents - Infinite slope
3. Discontinuities - Jumps or holes
If you can't draw a unique tangent line, the derivative doesn't exist.
The Absolute Value Function
Easy
Why isn't f(x) = |x| differentiable at x = 0?
1
Look at the graph
f(x) = |x| has a sharp corner at x = 0
2
Check left limit
From left: slope = -1
3
Check right limit
From right: slope = +1
4
Conclusion
Left and right slopes don't match → no unique tangent → not differentiable
Answer:
The left and right limits of the derivative are different
Key Takeaways
Differentiable means the derivative exists
Corners, cusps, vertical tangents, and discontinuities prevent differentiability
If differentiable, then continuous (but not vice versa!)