Master Continuity & Differentiability – Class 12 Maths (Chapter 5) | Theory, NCERT Solutions & CBSE PYQs

 Here’s the comprehensive guide for Chapter 5: Continuity & Differentiability (CBSE Class 12 Maths, 2025–26 syllabus):

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A. 🔑 Key Concepts and Theorems

1. Continuity

• A function $f$ is continuous at $x = a$ if $\lim_{x \to a}f(x) = f(a)$, meaning left and right limits equal the function value .

• Discontinuities:

  • Removable (hole), jump, and infinite/oscillatory .

• Algebra of continuous functions:

  • Sums, products, quotients (with non-zero denominator), and compositions of continuous functions are continuous .

• Intermediate Value Theorem / Rolle’s & Mean Value Theorem: continuous on [a,b] ⇒ there exist points satisfying f(b)=f(a) or f′(c) = (f(b)–f(a))/(b–a) .

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2. Differentiability

• $f$ is differentiable at $a$ if the limit $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists .

• Differentiable ⇒ Continuous, but not vice versa (e.g. $f(x)=|x|$ at 0) .

• Left & Right Derivatives: both must equal for differentiability.

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3. Core Derivative Rules

• Composite Functions (Chain Rule):
  $\frac{d}{dx} f(g(x)) = f′(g(x)) \cdot g′(x)$ .

• Implicit Differentiation: differentiate both sides of $F(x,y) = 0$, solve for $dy/dx$ .

• Logarithmic Differentiation: useful when $y = [u(x)]^{v(x)}$.

• Parametric and Second-order Derivatives: differentiate w.r.t parameter $t$, then find $d^2y/dx^2$ .

• Derivatives of exponential & logarithmic Functions: $d/dx(e^x)=e^x$, $d/dx(\ln x) = 1/x$ .

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B. 📘 NCERT Exercise Solutions

Structured by exercise:

• Ex 5.1–5.2: Continuity at a point/interval, verifying LHL = RHL = f(a).

• Ex 5.3–5.4: Checking differentiability; apply chain rule.

• Ex 5.5: Implicit differentiation problems.

• Ex 5.6: Logarithmic differentiation (e.g., $y = x^x$).

• Ex 5.7: Parametric and second-order derivatives.

• Ex 5.8: Verifications of Rolle’s and Mean Value Theorem.

Each solution:

1. State definitions/conditions.

2. Set up limits or derivatives.

3. Compute stepwise using standard rules.

4. Conclude clearly.

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C. 🏆 CBSE PYQs with Model Answers

2023

Q. Check continuity/differentiability of $f(x)=x|x|$ at 0.
A:

• Continuous: $\lim_{x→0^{-}} = \lim_{x→0^{+}} = 0 = f(0)$.

• Derivative left = $-0$, right = $+0$ ⇒ equal ⇒ differentiable.

2022

Q. Find $dy/dx$ if $\ln y = x \sin x$.
A:

$$\frac{1}{y} \cdot \frac{dy}{dx} = \sin x + x\cos x \Rightarrow \boxed{y(\sin x + x\cos x)}$$

2021

Q. If $x = \cos t, y = \sin t$, find $\frac{d^2y}{dx^2}$.
A:

$$\frac{dy}{dx} = \frac{\cos t}{- \sin t} = -\cot t; \quad \frac{d^2y}{dx^2} = \frac{1}{\sin^3 t}$$

2020

Q. Verify Rolle’s theorem for $f(x)=x^2-5x+6$ on [2,3].
A:

• $f(2)=0, f(3)=0$; continuous & differentiable; f′(x)=2x−5; f′(2.5)=0 ⇒ holds.

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#Tags:

#Class12Maths, #Continuity, #Differentiability, #ChainRule, #ImplicitDifferentiation, #MeanValueTheorem, #CBSEPYQs, #NCERTSolutions, #Calculus, #Chapter5Maths