After mastering this chapter, proceed to Applications of Derivatives, as it directly builds upon the concepts of continuity, differentiability, and derivative formulas. A strong understanding of this chapter makes optimization, tangents, normals, and monotonicity problems much easier.
Begin by revising the conditions for continuity, followed by all standard derivative formulas. Practice chain rule, implicit differentiation, logarithmic differentiation, and inverse trigonometric derivatives, then solve JEE Main & Advanced Previous Years' Questions (PYQs) to reinforce concepts.
Many students verify only the limit and forget to compare it with the actual function value. A function is continuous at a point only if the Left-Hand Limit (LHL), Right-Hand Limit (RHL), and the function value are all equal.
The highest-weightage topics include continuity of functions, differentiability, chain rule, implicit differentiation, logarithmic differentiation, and derivatives of inverse trigonometric, exponential, and logarithmic functions.
Continuity and Differentiability forms the foundation of Differential Calculus and is directly connected to Applications of Derivatives, Indefinite Integration, Definite Integration, Differential Equations, and Area Under Curves. It is one of the most frequently tested chapters in competitive examinations.
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Class 12 Maths Chapter 5 Continuity and Differentiability Revision Notes
From analyzing the motion of vehicles and optimizing engineering designs to understanding changing physical quantities in science and economics, Continuity and Differentiability form the foundation of Calculus. This chapter explains how functions behave without breaks and how their rates of change can be measured using derivatives. You'll learn concepts such as continuity, differentiability, derivative of composite functions, chain rule, implicit differentiation, logarithmic differentiation, and derivatives of inverse trigonometric functions, making this one of the highest-weightage chapters for JEE Main, JEE Advanced, and CBSE Board examinations.
At ALLEN, our expert faculty have designed these Class 12 Maths Chapter 5: Continuity and Differentiability Revision Notes to help you revise every concept quickly and effectively. These notes include concise theory, standard derivative formulas, shortcut techniques, solved illustrations, common mistakes, and PYQ-based insights. Whether you're preparing for CBSE Board exams or competitive exams like JEE Main and JEE Advanced, these revision notes will strengthen your Calculus fundamentals and improve your problem-solving speed.
1.0Chapter Snapshot
Topic
Overview
Continuity
Definition and continuity conditions
Differentiability
Concept and relationship with continuity
Derivatives of Composite Functions
Chain Rule
Implicit Differentiation
Differentiating implicit functions
Logarithmic Differentiation
Simplifying complex functions
Derivatives of Inverse Trigonometric Functions
Standard derivative formulas
Exponential & Logarithmic Functions
Derivative formulas and applications
Applications
Foundation for Differential Calculus
2.0Related Supporting Study Resources
Resource
Status
Class 12 Maths Chapter 5 NCERT Solutions
Available Soon
Continuity and Differentiability Important Questions
Available Soon
Formula Sheet PDF
Available Soon
JEE Main & Advanced Previous Year Questions
Available Soon
Chapter-wise Mock Test
Available Soon
Practice Worksheets
Available Soon
Quick Revision Notes PDF
Available Soon
Mind Maps
Available Soon
3.0Learning Outcomes
After completing these revision notes, you will be able to:
Understand the concepts of continuity and differentiability.
Verify whether a function is continuous or differentiable at a point.
Apply the chain rule to differentiate composite functions.
Solve problems using implicit and logarithmic differentiation.
Differentiate exponential, logarithmic, and inverse trigonometric functions.
Apply derivative formulas efficiently in problem-solving.
Solve JEE Main, JEE Advanced, and CBSE Board-level Calculus questions.
Build a strong foundation for Applications of Derivatives and Integration.
4.05-Minute Quick Revision
Must-Revise Concepts
Continuity
Differentiability
Left-Hand Limit (LHL)
Right-Hand Limit (RHL)
Derivative
Chain Rule
Implicit Differentiation
Logarithmic Differentiation
Derivatives of Inverse Trigonometric Functions
Exponential Functions
Logarithmic Functions
Important Formula Recall
Condition for Continuity:
limx→a−f(x)=limx→a+f(x)=f(a)
Chain Rule: dxdy=dudy⋅dxdu
Derivative of (e^x): dxd(ex)=ex
Derivative of (ln x): dxd(lnx)=x1
Derivative of (sin−1x): dxd(sin−1x)=1−x21
Derivative of (tan−1x): dxd(tan−1x)=1+x21
Important Properties
Every differentiable function is continuous, but every continuous function is not necessarily differentiable.
Continuity is verified by comparing LHL, RHL, and the function value.
Chain rule simplifies the differentiation of composite functions.
Logarithmic differentiation is useful for functions with variable exponents.
Implicit differentiation is applied when one variable cannot be expressed explicitly in terms of another.
5.0High Weightage Topics
Topic
JEE Importance
Difficulty Level
Continuity
Very High
Moderate
Differentiability
Very High
Moderate
Chain Rule
Very High
Moderate
Implicit Differentiation
High
Moderate
Logarithmic Differentiation
High
Moderate
Derivatives of Inverse Trigonometric Functions
Very High
Moderate
6.0Formula Handbook
The following formulas and standard results are essential for solving questions from Continuity and Differentiability in JEE Main, JEE Advanced, and CBSE examinations.
7.0Common Mistakes & JEE Tips
Continuity and Differentiability is one of the most important Calculus chapters in JEE. Students often lose marks by overlooking continuity conditions or applying differentiation rules incorrectly.
Remember that every differentiable function is continuous, but the converse is not always true.
Ignoring LHL and RHL while checking continuity
Verify LHL, RHL, and (f(a)) before concluding continuity.
Applying the chain rule incorrectly
Differentiate the outer function first, followed by the inner function.
Forgetting to use logarithmic differentiation
Apply logarithmic differentiation when variables appear in both the base and exponent.
Errors in implicit differentiation
Differentiate every term with respect to (x), including (y), using the chain rule.
Using incorrect derivative formulas for inverse trigonometric functions
Memorize all standard formulas along with their domains.
Ignoring domain restrictions
Check the domain before evaluating derivatives or limits.
Sign errors while differentiating
Write every differentiation step carefully to avoid losing marks.
8.0ALLEN Faculty Tips
Memorize all standard derivative formulas before solving advanced problems.
Revise the relationship between continuity and differentiability regularly.
Practice chain rule, implicit differentiation, and logarithmic differentiation every day.
Learn the derivative formulas of inverse trigonometric functions thoroughly.
Solve graphical questions to understand continuity and differentiability visually.
Practice Previous Years' Questions (PYQs) to improve speed and conceptual clarity.
Focus on understanding concepts instead of memorizing lengthy solutions.
9.0PYQ Trend Analysis
Continuity and Differentiability is one of the highest-weightage chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Questions are commonly asked on continuity, differentiability, chain rule, implicit differentiation, logarithmic differentiation, and derivatives of inverse trigonometric functions.
Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.
Topic
Questions Asked (Last 5 Years)
Average Difficulty
Continuity
2–3
Moderate
Differentiability
2–3
Moderate
Chain Rule
2–3
Moderate
Implicit Differentiation
1–2
Moderate
Logarithmic Differentiation
1–2
Moderate
Derivatives of Inverse Trigonometric Functions
2–3
Moderate
10.0Most Common Question Types
Checking whether a function is continuous at a given point.
Determining the differentiability of piecewise-defined functions.
Applying the chain rule to differentiate composite functions.
Solving implicit differentiation problems.
Using logarithmic differentiation for complex expressions.
Finding derivatives of exponential, logarithmic, and inverse trigonometric functions.
Evaluating limits involving continuity and differentiability.