Worksheets · Calculus

Differentiability worksheet

A function is differentiable at a point when its graph has one clear, non-vertical tangent there. A jump breaks continuity first; a corner or cusp gives two different slopes; a vertical tangent gives a slope that is infinite.

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Name

Differentiability

Date Period

Decide whether each function is differentiable at the given point. If not, say why.

  1. Write ff in pieces if it helps. Is it differentiable at the point? If not, name the reason

    Is f(x)=x+33+4x−6 differentiable at x=−3?
  2. Is f(x)=x∣x+3∣+3 differentiable at x=−3?
  3. Is f(x)=x2+4x+43−3 differentiable at x=−2?
  4. Is f(x)=∣x−2∣x−2−6 differentiable at x=2?
  5. Is f(x)=∣x∣+∣x−5∣−5 differentiable at x=3?
  6. Is f(x)=x2−25∣x−5∣+6 differentiable at x=5?
  7. Is f(x)=(x+5)∣x+5∣+5 differentiable at x=−5?
  8. Is f(x)=x2+4x+4−2 differentiable at x=−2?
  9. Is f(x)=x2−6x+93−8 differentiable at x=3?
  10. Is f(x)=∣x+4∣+∣x+2∣−1 differentiable at x=−4?

  1. Check continuity first, then the one-sided derivatives

    Is f(x)={−x−1x≤−1x+1x>−1 differentiable at x=−1?
  2. Is f(x)={x2+2x−1x≤22x2−2x+3x>2 differentiable at x=2?
  3. Is f(x)={−x2−x−2x<1−1x=1−2x−2x>1 differentiable at x=1?
  4. Is f(x)={x3−2x−4x≤−210x+10x>−2 differentiable at x=−2?
  5. Is f(x)={−3x+5x<3−2x2+7x−9x≥3 differentiable at x=3?
  6. Is f(x)={x3−x+1x≤−2−x2+10x+19x>−2 differentiable at x=−2?
  7. Is f(x)={x+3x<−1−1x=−1x2+x+2x>−1 differentiable at x=−1?
  8. Is f(x)={x2−3x<−32x2+8x+14x≥−3 differentiable at x=−3?
  9. Is f(x)={2x2+2x+4x<210x−4x≥2 differentiable at x=2?
  10. Is f(x)={x−3x<1−5x=12x−4x>1 differentiable at x=1?

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

DifferentiabilityAnswer keyVersion 1

Date Period

Decide whether each function is differentiable at the given point. If not, say why.

  1. Write ff in pieces if it helps. Is it differentiable at the point? If not, name the reason

    Is f(x)=x+33+4x−6 differentiable at x=−3?No: a vertical tangent
  2. Is f(x)=x∣x+3∣+3 differentiable at x=−3?No: a corner
  3. Is f(x)=x2+4x+43−3 differentiable at x=−2?No: a cusp
  4. Is f(x)=∣x−2∣x−2−6 differentiable at x=2?No: not continuous (a jump)
  5. Is f(x)=∣x∣+∣x−5∣−5 differentiable at x=3?Yes
  6. Is f(x)=x2−25∣x−5∣+6 differentiable at x=5?No: not continuous (a jump)
  7. Is f(x)=(x+5)∣x+5∣+5 differentiable at x=−5?Yes
  8. Is f(x)=x2+4x+4−2 differentiable at x=−2?No: a corner
  9. Is f(x)=x2−6x+93−8 differentiable at x=3?No: a cusp
  10. Is f(x)=∣x+4∣+∣x+2∣−1 differentiable at x=−4?No: a corner

  1. Check continuity first, then the one-sided derivatives

    Is f(x)={−x−1x≤−1x+1x>−1 differentiable at x=−1?No, a corner. Both pieces give 0, but f′=−1 from the left and 1 from the right.
  2. Is f(x)={x2+2x−1x≤22x2−2x+3x>2 differentiable at x=2?Yes. Both pieces give 7, and f′=6 from both sides.
  3. Is f(x)={−x2−x−2x<1−1x=1−2x−2x>1 differentiable at x=1?No: not continuous. Both pieces approach −4, but f(1)=−1.
  4. Is f(x)={x3−2x−4x≤−210x+10x>−2 differentiable at x=−2?No: not continuous. The left piece gives −8, the right piece −10.
  5. Is f(x)={−3x+5x<3−2x2+7x−9x≥3 differentiable at x=3?No: not continuous. The left piece gives −4, the right piece −6.
  6. Is f(x)={x3−x+1x≤−2−x2+10x+19x>−2 differentiable at x=−2?No, a corner. Both pieces give −5, but f′=11 from the left and 14 from the right.
  7. Is f(x)={x+3x<−1−1x=−1x2+x+2x>−1 differentiable at x=−1?No: not continuous. Both pieces approach 2, but f(−1)=−1.
  8. Is f(x)={x2−3x<−32x2+8x+14x≥−3 differentiable at x=−3?No: not continuous. The left piece gives 6, the right piece 8.
  9. Is f(x)={2x2+2x+4x<210x−4x≥2 differentiable at x=2?Yes. Both pieces give 16, and f′=10 from both sides.
  10. Is f(x)={x−3x<1−5x=12x−4x>1 differentiable at x=1?No: not continuous. Both pieces approach −2, but f(1)=−5.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Is f(x)=∣x−2∣ differentiable at x=2?

  1. To the left of 2 the graph has slope -1; to the right it has slope 1.
  2. The two one-sided slopes disagree, so there is no single tangent line.

The answer is No: a corner.

Where students go wrong

Deciding from continuity alone. Every differentiable function is continuous, but |x| is continuous at 0 and still not differentiable there.

Also called is f differentiable, differentiable vs continuous or where is f not differentiable.

What you can put on this worksheet

Differentiable at a point, and why
Is f(x)=x−53−5 differentiable at x=6? → Yes
Graphs and piecewise functions
Use the graph of r. Is r differentiable at 4, at −1 and at −7? If not, say why. → x=4: no, a vertical tangent; x=−1: no, not continuous (a hole); x=−7: yes

Questions about these worksheets

Yes. Every differentiability sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the differentiability questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 2: differentiable at a point, and why and graphs and piecewise functions. Tick as many as you want and set how many of each, or let it spread them evenly.

Deciding from continuity alone. Every differentiable function is continuous, but |x| is continuous at 0 and still not differentiable there.