A derivative exists at a point only if the defining limit exists there. Several ordinary graphs fail that test.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Corners
At the corner of y = |x| the left slope is −1 and the right slope is 1. They disagree, so no tangent exists.
Vertical tangents and cusps
A vertical tangent has undefined slope, and a cusp is where the two sides run to opposite infinities.
Differentiable implies continuous
A break makes the defining limit fail immediately, so every differentiable function is continuous.
The converse is false
y = |x| is continuous everywhere and not differentiable at zero. Continuity is necessary, not sufficient.
Local linearity
Differentiable means locally linear: zoom in far enough and the curve becomes indistinguishable from its tangent.
Where a derivative fails to exist
Corners, cusps, vertical tangents and discontinuities. Each has a distinct visual signature, and identifying which one applies is what an exam justification requires.
A corner has two one-sided derivatives
At the corner of |x| the derivative from the left is −1 and from the right +1. They disagree, so no derivative exists — even though the function is perfectly continuous there.
Local linearity is what differentiability means
Zoom in on a differentiable point and the curve becomes indistinguishable from its tangent line. That is the geometric content of differentiability and the justification for linear approximation.
Piecewise differentiability
For a piecewise function to be differentiable at a boundary, the pieces must match in value and in slope. That gives two equations, which is why these problems usually have two unknown constants.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Diego Test a Corner
Diego checks whether y = |x − 3| is differentiable at x = 3.
- Step 1
To the left of 3 the function is 3 − x, so the slope there is −1.
Step 4: Your Turn
Practice makes it stick.
The Corner
Problem 1 of 2
y = |x − 3|. At which x is it not differentiable?
The Implication
Problem 2 of 2
A function is differentiable at x = 5. Must it be continuous there? 1 yes, 0 no.
Where the Slope Fails
1 of 8
y = |x|. Slope just to the right of zero?
2 of 8
y = |x|. Slope just to the left of zero?
3 of 8
Continuous at x = 2. Must it be differentiable? 1 yes, 0 no.
4 of 8
y = |x + 7|. At which x is it not differentiable?
5 of 8
A graph jumps at x = 1. Differentiable there? 1 yes, 0 no.
6 of 8
At a vertical tangent, is the derivative defined? 1 yes, 0 no.
7 of 8
Which situations block differentiability?
8 of 8
y = x². Differentiable at zero? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = |x − 5|. At which x is the derivative undefined?
Question 2 of 2
What does local linearity mean?
What You Learned
- A derivative fails at corners, cusps, vertical tangents and breaks.
- Differentiable implies continuous, but not the reverse.
- Differentiable means locally linear.