JEE/Mathematics/Differentiability

Calculus · Must Do

Differentiability — JEE Main & Advanced Notes

Differentiability measures smooth local change and automatically implies continuity at the point.

derivativeLHD/RHDchain ruleimplicit
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1. Introduction & Exam Weightage

Differentiability measures smooth local change and automatically implies continuity at the point.

Material signal: Mapped from continuity-differentiability files.

Priority: Must Do. Treat this as a moderate chapter inside the JEE Mathematics ladder.

2. Core Concepts & Definitions

  • Differentiability needs equal left and right derivatives.
  • Corners, cusps and vertical tangents often fail differentiability.
  • Chain rule is function-inside-function rate transfer.

3. Key Formulas with Derivation Hints

  • f'(a)=lim_{h o0} rac{f(a+h)-f(a)}h
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
  • (uv)'=u'v+uv'
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
  • (f(g(x)))'=f'(g(x))g'(x)
    Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
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4. Solved Examples — Original Practice Models

Differentiability: main-style warm-up

Problem: A direct one-step Differentiability question is solved by identifying the governing definition before substituting values.

Method: write the target quantity, list the known values, use the standard relation, and check if the answer is dimensionally or logically possible. This prevents option-led guessing.

Differentiability: advanced-style trap

Problem: A multi-condition Differentiability problem often has an extra restriction hidden in domain, sign, interval or geometry.

Method: solve algebraically first, then filter using the domain/interval/geometry condition. In JEE Advanced, most wrong options come from skipping this final filtering step.

5. Common Mistakes & Traps

  • Assuming continuity implies differentiability.
  • Forgetting product/chain rules.
  • Mishandling modulus at zero points.

6. JEE Main Specific Strategy

For JEE Main, aim for fast recognition and clean substitution. Finish the first pass of Differentiability questions in 60–90 seconds each. Prioritise standard formulas, short sign/domain checks and option elimination only after the setup is correct.

7. JEE Advanced Specific Strategy

For JEE Advanced, expect combined conditions, hidden domains, multi-correct traps and integer-style answers. Build the solution from definitions, not memorised tricks. When a parameter appears, solve the general case and then filter using restrictions.

8. Quick Revision Summary

  • Differentiable => continuous.
  • Check LHD/RHD for piecewise/modulus.
  • Use logarithmic differentiation for products with powers.
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