CBSE · Senior secondary

CBSE Class 12 — Notes, Chapters & Practice Quizzes

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Mathematics · 13 chapters
Summary, key terms, extra questions with answers and a practice quiz with AI diagnosis for each.

Chapter 5: Continuity and DifferentiabilityClass 12 Mathematics — summary, notes, extra questions & MCQ quiz

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\(f\) is continuous at \(x=c\) if:

Summary

This chapter formalises continuity and differentiability. A function \(f\) is continuous at \(x=c\) if \(\lim_{x\to c}f(x)=f(c)\); it is continuous on an interval if continuous at every point. Differentiability is stronger: \(f'(c)\) exists if the limit defining the derivative exists, and differentiability implies continuity (but not conversely). The chapter develops rules for differentiating composite functions (chain rule), implicit functions, inverse trigonometric functions, exponential and logarithmic functions, and functions given in parametric form. Logarithmic differentiation handles products, quotients and variable exponents. Second-order derivatives \(\dfrac{d^2y}{dx^2}\) are introduced, and the chapter closes with the Mean Value Theorems of Rolle and Lagrange, which connect the derivative to the behaviour of the function on an interval.

Continuity of functionsDifferentiability and the chain ruleDerivatives of inverse trig, exponential and log functionsLogarithmic and parametric differentiationMean Value Theorems

Key terms

Continuity at a point
\(\lim_{x\to c}f(x)=f(c)\).
Differentiability
Existence of \(f'(c)=\lim_{h\to0}\dfrac{f(c+h)-f(c)}{h}\).
Chain rule
\(\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}\) for composite functions.
Implicit differentiation
Differentiating an equation in \(x\) and \(y\) without solving for \(y\).
Logarithmic differentiation
Taking \(\log\) before differentiating, useful for \(y=u^v\).
Mean Value Theorem
There is \(c\) with \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\) (Lagrange).

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Practice quiz · Continuity and Differentiability

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