Class 12 Maths NCERT Solutions Chapter 5 Continuity and Differentiability | Boundless Maths
Unit III · Calculus · Chapter 5

Class 12 Maths NCERT Solutions Chapter 5: Continuity and Differentiability

Free, step-by-step NCERT Solutions for all seven exercises of this chapter — continuity, derivatives of composite and implicit functions, derivatives of inverse trigonometric, exponential and logarithmic functions, logarithmic differentiation, parametric forms, and second order derivatives. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

This is the chapter that takes the differentiation you learned in Class 11 and makes it rigorous — and it's the one every later Calculus chapter leans on. Get the chain rule and implicit differentiation solid here, and Application of Derivatives, Integrals, and Differential Equations all get noticeably easier to pick up.

8Exercises (incl. Misc.)
137Total Questions
2026-27CBSE Syllabus
100%Solved

Class 12 Maths NCERT Solutions Chapter 5 — Overview

This Class 12 Maths NCERT Solutions Chapter 5 hub covers Continuity and Differentiability, which picks up the differentiation you learned in Class 11 and makes it rigorous — and far more powerful. You'll first pin down exactly what it means for a function to be continuous at a point, then use that idea to define differentiability precisely: a function is differentiable where its left hand and right hand derivatives exist and agree. Every differentiable function turns out to be continuous, but the reverse isn't true, and the classic counterexample — f(x) = |x| at x = 0 — shows up constantly in exam questions.

From there, the chapter builds the entire differentiation toolkit you'll rely on for the rest of Calculus: the chain rule for composite functions, implicit differentiation for relations that can't easily be solved for y, derivatives of inverse trigonometric functions, a new class of exponential and logarithmic functions with their own clean derivatives, logarithmic differentiation for expressions of the form [u(x)]^v(x), functions defined parametrically through a third variable, and finally second order derivatives. Application of Derivatives (Chapter 6) depends on every one of these techniques, so this is the chapter to get airtight.

How the Chapter Builds

One Idea Leads to the Next

1

Continuity

No break in the graph: the limit at c equals f(c). Exercise 5.1.

→
2

Differentiability & Chain Rule

Differentiable ⟹ continuous. Composite functions differentiate via the chain rule. Exercise 5.2.

→
3

Implicit & Special Functions

Differentiate relations directly; derivatives of inverse trig, exponential and log functions. Exercises 5.3–5.4.

→
4

Logarithmic Differentiation

Handle [u(x)]^v(x) by taking log of both sides first. Exercise 5.5.

→
5

Parametric & Second Order

x and y linked via a parameter t; then differentiate again for d²y/dx². Exercises 5.6–5.7.

Quick Reference

Important Formulas — Chapter 5

Everything you need before you start solving. This is a summary for quick recall — the Formula Deck below has the full printable version for all of Calculus.

Continuity (§5.2)

Continuity at a point c

\lim_{x \to c} f(x) = f(c)

Left hand limit, right hand limit and f(c) must all exist and be equal.

Algebra of continuous functions

f \pm g,\; f \cdot g,\; \dfrac{f}{g}\; (g(c)\neq 0)

All continuous at c whenever f and g are continuous at c.

Composite of continuous functions

If g is continuous at c and f is continuous at g(c), then f\circ g is continuous at c.

Useful for functions like sin(x²) or |1 − x + |x||.

Differentiability & Chain Rule (§5.3)

Derivative at c

f'(c) = \lim_{h \to 0} \dfrac{f(c+h)-f(c)}{h}

Exists only if left hand and right hand derivatives are finite and equal.

Differentiable ⟹ Continuous

Every differentiable function is continuous — but the converse is false (e.g. f(x) = |x| at x = 0).

A near-guaranteed 1-mark or reasoning question every year.

Chain Rule

\dfrac{df}{dx} = \dfrac{dv}{dt}\cdot\dfrac{dt}{dx}, \quad f = v \circ u,\; t = u(x)

The single most-used rule in this chapter and every chapter after it.

Implicit & Inverse Trigonometric Derivatives (§5.3)

Implicit differentiation

Differentiate both sides of the relation w.r.t. x directly, applying the chain rule to every term containing y.

Used when y cannot easily be isolated, e.g. x + sin xy − y = 0.

sin⁻¹x, cos⁻¹x

\dfrac{d}{dx}(\sin^{-1}x)=\dfrac{1}{\sqrt{1-x^2}},\quad \dfrac{d}{dx}(\cos^{-1}x)=\dfrac{-1}{\sqrt{1-x^2}}

Defined only for x ∈ (−1, 1).

tan⁻¹x

\dfrac{d}{dx}(\tan^{-1}x)=\dfrac{1}{1+x^2}

Defined for all real x.

Exponential, Logarithmic & Logarithmic Differentiation (§5.4–5.5)

Natural exponential & log

\dfrac{d}{dx}(e^x)=e^x, \qquad \dfrac{d}{dx}(\log x)=\dfrac{1}{x}

eˣ is the only function that doesn't change under differentiation.

General exponential

\dfrac{d}{dx}(a^x) = a^x \log a, \quad a>0

Derived by writing a^x = e^{x \log a} and applying the chain rule.

Logarithmic differentiation

For y=[u(x)]^{v(x)}, take log of both sides first: \log y = v(x)\log[u(x)], then differentiate.

Required whenever the variable appears in both the base and the exponent. f(x) and u(x) must be positive.

Parametric Forms & Second Order Derivatives (§5.6–5.7)

Parametric derivative

\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}, \quad \dfrac{dx}{dt}\neq 0

x = f(t), y = g(t) — differentiate each w.r.t. t, then divide.

Second order derivative

\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)

Also written f″(x), D²y, y″ or y₂. Differentiate the first derivative again.

Decision Guide

Which Differentiation Technique Should I Use?

A quick way to decide once you look at how x and y are related.

SituationUse thisWhy
y = f(x) is a composite of two or more functions, e.g. sin(x²)Chain RuleDifferentiate the outer function, keep the inner function unchanged, then multiply by the inner function's derivative.
The relation between x and y can't easily be solved for y, e.g. x + sin xy − y = 0Implicit DifferentiationDifferentiate both sides directly w.r.t. x, treating y as a function of x via the chain rule.
The variable appears in both the base and the exponent, e.g. xˢⁱⁿˣ or (log x)ˣLogarithmic DifferentiationOrdinary power and exponential rules don't apply here — taking log first turns the exponent into a product.
x and y are both given in terms of a third variable tParametric DifferentiationFind dy/dt and dx/dt separately, then divide — never eliminate the parameter unless asked.
The question asks for d²y/dx² or asks you to "prove" a relation involving derivativesSecond Order DerivativeDifferentiate dy/dx once more w.r.t. x, simplifying carefully before substituting.
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across all seven exercises.

  • Assuming continuous means differentiable. f(x) = |x| is continuous everywhere but not differentiable at x = 0 — always check left hand and right hand derivatives separately at a suspected corner point.
  • Forgetting to check left and right hand limits separately for piecewise functions at the point where the definition changes — checking only one side is a common source of lost marks.
  • Skipping the chain rule when differentiating composite functions — writing d/dx(sin x²) = cos x² instead of 2x cos x².
  • Forgetting dy/dx on the y-terms during implicit differentiation — every term containing y needs a trailing dy/dx from the chain rule.
  • Ignoring domain restrictions on inverse trig derivatives — the formulas for sin⁻¹x and cos⁻¹x only hold for x ∈ (−1, 1); forgetting this leads to sign errors.
  • Attempting logarithmic differentiation when the base or the whole expression can be negative — logs of negative numbers aren't defined, so the technique only applies where f(x) and u(x) are positive.
  • Eliminating the parameter unnecessarily in parametric problems — dy/dx is usually expected purely in terms of the parameter, not x and y.
  • Not simplifying dy/dx before finding d²y/dx² — differentiating an unsimplified first derivative makes the second order derivative far messier and more error-prone than it needs to be.
Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 5 — Choose an Exercise

5.1

Exercise 5.1

Continuity — checking continuity at a point, finding points of discontinuity, finding unknown constants · 34 questions

Solve Exercise 5.1 →
5.2

Exercise 5.2

Derivatives of composite functions using the chain rule, and non-differentiability at a point · 10 questions

Solve Exercise 5.2 →
5.3

Exercise 5.3

Derivatives of implicit functions and inverse trigonometric functions · 15 questions

Solve Exercise 5.3 →
5.4

Exercise 5.4

Derivatives of exponential and logarithmic functions · 10 questions

Solve Exercise 5.4 →
5.5

Exercise 5.5

Logarithmic differentiation — functions of the form [u(x)]^v(x) and related product/quotient forms · 18 questions

Solve Exercise 5.5 →
5.6

Exercise 5.6

Derivatives of functions given in parametric form, without eliminating the parameter · 11 questions

Solve Exercise 5.6 →
5.7

Exercise 5.7

Second order derivatives, including proofs of differential-equation-style relations · 17 questions

Solve Exercise 5.7 →
M

Miscellaneous Exercise

Mixed application questions combining every technique from the chapter · 22 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Calculus — continuity, differentiability, applications of derivatives, integration — in one printable PDF.

Get Formula Deck →

🤖 Want Board Exam-Style Practice?

The AI Question Bank has board-tagged MCQs, Assertion-Reason and Case Studies for Continuity and Differentiability, with instant feedback.

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Common Questions

Frequently Asked Questions

Quick answers from Class 12 Maths NCERT Solutions Chapter 5, Continuity and Differentiability.

How many exercises are there in Chapter 5, Continuity and Differentiability?
There are seven main exercises — 5.1 through 5.7 — plus a Miscellaneous Exercise, totalling 137 questions across continuity, differentiability, chain rule, implicit and logarithmic differentiation, inverse trigonometric and exponential/log derivatives, parametric forms, and second order derivatives.
Is this chapter important for the board exam?
Yes — it's the foundation of the entire Calculus unit, and every later chapter (Application of Derivatives, Integrals, Differential Equations) depends directly on the differentiation techniques built here, especially the chain rule and implicit differentiation.
What is the difference between continuity and differentiability?
A function is continuous at a point if its graph has no break there — the limit equals the function value. Differentiability is a stronger condition: it requires the left hand and right hand derivatives to exist and be equal. Every differentiable function is continuous, but a continuous function need not be differentiable — f(x) = |x| is continuous at x = 0 but not differentiable there.
What should I revise before starting this chapter?
Make sure your Class 11 basics of limits and differentiation of polynomial and trigonometric functions are solid — this chapter builds directly on them before introducing continuity, the chain rule, and several new classes of functions.
Where can I find the official NCERT textbook for this chapter?
Continuity and Differentiability is Chapter 5 of the NCERT Class 12 Mathematics textbook (Part I), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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