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Differentiability questions in JEE look different each time: a modulus, a greatest integer function, a piecewise function with unknown constants. They all use one test. This guide gives you that test, applies it to each common type with worked numbers, and lists the mistakes that cost marks.
What does differentiable at a point mean?
The NCERT Class 12 chapter Continuity and Differentiability defines it through the limit of [f(c + h) − f(c)] ÷ h as h → 0. In practice you compute two one-sided limits:
Left-hand derivative (LHD): h → 0 from below.
Right-hand derivative (RHD): h → 0 from above.
The function is differentiable at c when both exist, are finite and are equal.
NCERT proves that a differentiable function is continuous, and shows that the converse fails with f(x) = |x|. At 0 the LHD is −1 and the RHD is 1, so |x| is continuous but not differentiable there. That one example is the template for most questions.
How do you find the corners of a modulus function?
A function |g(x)| can fail to be differentiable where g(x) = 0, because the graph folds up there.
Example. Where is f(x) = |x² − 3x + 2| not differentiable?
Factor: x² − 3x + 2 = (x − 1)(x − 2). Zeros at x = 1 and x = 2.
The derivative of the inside is g′(x) = 2x − 3. At x = 1, g′ = −1. At x = 2, g′ = 1. Neither is zero.
At each zero the LHD and RHD are −g′ and +g′ in some order, so they differ.
Answer: not differentiable at x = 1 and x = 2.
The rule: |g(x)| is not differentiable at a zero of g where g′ is non-zero. If g′ is also zero there, check directly. For example, |x³| and x|x| are both differentiable at 0.
Check x|x| at 0. [f(h) − f(0)] ÷ h = h|h| ÷ h = |h|, which tends to 0 from both sides. Derivative exists and equals 0.
How do you solve piecewise questions with unknown constants?
Example. f(x) = x² for x ≤ 1, and f(x) = ax + b for x > 1. Find a and b so that f is differentiable at x = 1.
Continuity at 1. Left value: 1² = 1. Right limit: a + b. So a + b = 1.
Derivatives. LHD = 2x at x = 1, which is 2. RHD = a. So a = 2.
Then b = 1 − 2 = −1.
Check: the right piece is y = 2x − 1, which is exactly the tangent line to y = x² at (1, 1). The graph joins without a corner.
Why step 1 matters. Take f(x) = x for x ≤ 0 and f(x) = x + 1 for x > 0. Both pieces have slope 1, so comparing slopes alone suggests "differentiable". But the function jumps from 0 to 1 at x = 0, so it is not even continuous. Not differentiable.
What other function types come up?
Greatest integer function. NCERT asks you to prove that [x] is not differentiable at x = 1 and x = 2 for 0 < x < 3. It is not continuous at integers, so it fails at once.
Oscillating functions. For f(x) = x sin(1/x) with f(0) = 0, the difference quotient at 0 is sin(1/h), which has no limit, so f is not differentiable at 0, though it is continuous. For f(x) = x² sin(1/x) with f(0) = 0, the quotient is h sin(1/h), which tends to 0, so f′(0) = 0.
Sums of moduli. f(x) = |x − 1| + |x − 3| has corners at x = 1 and x = 3.
Why does the graph picture mislead?
Many students hold a picture of differentiable as "a smooth curve" and continuous as "drawn without lifting the pen". Tall and Vinner (1981) called this a concept image: the set of pictures and examples in a student's mind, which can differ from the formal definition and cause conflict. Their work focused on limits and continuity, but the lesson carries over. When a question is unusual, return to the LHD and RHD definition rather than your mental picture. Our post on limits and continuity covers the groundwork.
What are the common mistakes?
Comparing slopes without checking continuity. See the jump example above.
Calling every zero of g a corner of |g|. It is a corner only where g′ is non-zero.
Forgetting the domain. A one-sided derivative at an endpoint is judged differently from an interior point.
Differentiating the formula and substituting. For x² sin(1/x), the derivative formula for x ≠ 0 has no limit at 0, yet f′(0) exists. Use the definition at the special point.
Where is this in the syllabus?
The JEE (Main) 2026 syllabus lists "limits, continuity and differentiability" in Unit 7, and the CBSE Mathematics 2026-27 curriculum lists "continuity and differentiability" in Class XII calculus. Check the current documents for your exam year.
A practice routine
Day 1: Five modulus functions. Find zeros of the inside, check g′ at each, then confirm with LHD and RHD.
Day 2: Five piecewise questions with constants. Write "continuity" and "derivative" as separate lines every time.
Day 3: Greatest integer and oscillating functions from NCERT and previous papers.
Day 5: A mixed set of ten, timed. Mark each error by type.
If you keep choosing the wrong method, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Start with |x² − 5x + 6| today: find its corners and confirm one with the left and right derivatives.
