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Continuity and Differentiability

NCERT Class 12 Mathematics Chapter 5

Continuity and Differentiability for Class 12 Maths and JEE: continuity, the chain rule, implicit, exponential, logarithmic and second order derivatives.

6 topics Updated 8 Nov 2026

For: Class 12 and JEE

In the syllabus: JEE Class 12 syllabus

With your textbook

NCERT Mathematics Part I (Class 12), Chapter 5 “Continuity and Differentiability”, reprint 2026-27.

Open the official NCERT chapter (PDF, 3.2 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Continuity of a function

    Continuity of a function for Class 12 Maths and JEE: continuity at a point, one-sided limits, types of discontinuity, sums and composites, solved examples.

    NCERT Section 5.2 14 min read 5 solved examples 11 practice questions

  2. Differentiability and the chain rule

    Differentiability for Class 12 Maths and JEE: why |x| fails at 0, continuity vs differentiability, the chain rule, implicit and inverse trig derivatives.

    NCERT Section 5.3 16 min read 5 solved examples 13 practice questions

  3. Exponential and logarithmic functions

    Exponential and logarithmic functions for Class 12 Maths and JEE: graphs, laws of logarithms, change of base, derivatives of e and log x, solved examples.

    NCERT Section 5.4 12 min read 4 solved examples 9 practice questions

  4. Logarithmic differentiation

    Logarithmic differentiation for Class 12 Maths and JEE: formula for uv, the derivative of xx, products and quotients, sums of powers, solved examples.

    NCERT Section 5.5 15 min read 5 solved examples 10 practice questions

  5. Derivatives of functions in parametric form

    Parametric differentiation for Class 12 Maths and JEE: derivatives in parametric form, dy/dx = (dy/dt)/(dx/dt), circles, parabolas, solved examples.

    NCERT Section 5.6 12 min read 5 solved examples 8 practice questions

  6. Second order derivative

    Second order derivatives for Class 12 Maths and JEE: dy/dx, y″ and y, relations like y − 5y + 6y = 0, implicit and parametric cases, solved examples.

    NCERT Section 5.7 14 min read 5 solved examples 8 practice questions

Key formulas in this chapter

Continuity

  • at an interior point c: f continuous ⇔ limx→c− f(x) = limx→c+ f(x) = f(c)
  • f + g, f − g, fg, f/g (g(c) ≠ 0) continuous at c
  • g continuous at c, f continuous at g(c) ⇒ f∘g continuous at c

Differentiability

  • f differentiable at c ⇒ f continuous at c
  • df/dx = (dv/dt)(dt/dx), for f = v∘u and t = u(x)
  • d/dx (sin x) = 1/√(1 − x); d/dx (cos x) = −1/√(1 − x), for |x| < 1
  • d/dx (tan x) = 1/(1 + x)

Exponential and log functions

  • logb (pq) = logb p + logb q; logb (x/y) = logb x − logb y
  • loga p = logb p / logb a, for p > 0, a,b > 0 and a,b ≠ 1
  • d/dx (ex) = ex; d/dx (log x) = 1/x (x > 0)

Logarithmic differentiation

  • dy/dx = y[v u′/u + v′ log u], for y = uv, u > 0
  • d/dx (ax) = ax log a (a > 0)
  • d/dx (xx) = xx(1 + log x) (x > 0)

Parametric derivatives

  • dy/dx = (dy/dt)/(dx/dt), for dx/dt ≠ 0
  • x = a cos θ, y = a sin θ: dy/dx = −cot θ (a > 0, sin θ ≠ 0)
  • x = at, y = 2at: dy/dx = 1/t (a > 0, t ≠ 0)

Second order derivative

  • dy/dx = d/dx (dy/dx)
  • y = A sin x + B cos x ⇒ y″ + y = 0
  • y = sin x ⇒ (1 − x)y″ − xy′ = 0 (|x| < 1)

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