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Principle deep dive

Differentiability at a point

Differentiability ⇒ continuity (not the converse). Modulus and greatest-integer are the standard counter-examples. Spans Limits & Continuity, Differentiation, and Functions.

questions in the bank
24
tagged HARD
17%
chapter spread
2
worked examples below
4

When to reach for it

Asked whether a function (often piecewise, |x|, or [x]) is differentiable at a point.

Why this principle matters

f is differentiable at x = c iff the left and right derivatives exist AND are equal. The left derivative is lim h→0⁻ [f(c + h) − f(c)] / h, the right is lim h→0⁺ [f(c + h) − f(c)] / h.

|x| is the canonical counter-example to 'continuous ⇒ differentiable'. At x = 0, the left derivative is −1, the right derivative is +1. Continuous (left limit = right limit = f(0) = 0), but not differentiable (slopes differ).

[x] (greatest integer function) is worse: at every integer, [x] jumps by 1. The derivative is 0 on every open interval (n, n+1), but undefined at integers. NDA tests this in derivative-of-floor questions — the answer is almost always 0 on the interior, undefined at the endpoint.

4 worked examples from the bank

Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.

Example 1DifferentiationHARD
Let f(x)=axx+1+b, x<1f(x)=\frac{ax}{x+1}+b,\ x<1 and x−1, 1≤x≤2\sqrt{x-1},\ 1\leq x\leq2.
If f(x)f(x) is differentiable at x=1x=1, then what is the value of (a+b)(a+b)?

[Q63 · Sep · 2023]

Example 2DifferentiationEASY
If f(x)=e∣x∣f(x)=e^{|x|}, then which one of the following is correct?

[Q84 · Apr · 2021]

Example 3DifferentiationEASY
Let y=[x+1]y=[x+1], −4<x<−3-4<x<-3 where [.][.] is the greatest integer function. What is the derivative of yy with respect to xx at x=−3.5x=-3.5?

[Q71 · Apr · 2022]

Example 4DifferentiationMODERATE
For the following three (03) items: Consider the function f(x)=x∣x∣f(x)=x|x|.
Consider the following statements: I. The function is increasing in the interval (−∞,∞)(-\infty,\infty). II. The function is differentiable at x=0x=0. Which of the statements given above is/are correct?

[Q75 · Sep · 2025]

Variants to recognise

Same principle, different surfaces. Pattern-match these on test day.

  • Left derivative = Right derivative

    Test both sides separately. If left and right slopes differ, f is not differentiable at the point.

  • Non-differentiability of |x| at 0

    Sharpest corner — slopes are −1 and +1. f(x) = |x − a| is non-differentiable at x = a.

  • [x] derivative on (n, n+1)

    Equals 0 on the open interval (constant in between). Undefined at integers (jump discontinuity).

  • Composition: e^|x|, ln |x|, etc.

    Often differentiable everywhere EXCEPT at the modulus's split point. Chain rule confirms.

Drill every differentiability at a point question

24 questions from the bank — paginated, with cart and Word-export support.

Drill the 24 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.