Continuity and Differentiability

part 1

Continuity

• Continuity at a point
• Continuity on an interval
• Algebra of continuous functions
• Continuity of composite functions

part 2

Differentiability and Standard Derivatives

• Differentiability and relation with continuity
• Derivatives of composite functions — chain rule
• Derivatives of implicit functions
• Derivatives of inverse trig functions
• Derivatives of exponential and logarithmic functions

part 3

Advanced Differentiation

• Logarithmic differentiation
• Derivatives of functions in parametric form
• Second order derivatives
• Rolle's theorem and Mean Value Theorem

Chain rule

$\frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x)$

★ board favourite

d/dx(sin⁻¹x)

$\frac{1}{\sqrt{1-x^2}}$

★ board favourite

d/dx(cos⁻¹x)

$-\frac{1}{\sqrt{1-x^2}}$

★ board favourite

d/dx(tan⁻¹x)

$\frac{1}{1+x^2}$

★ board favourite

Rolle's theorem

$f(a)=f(b)$, f cont., f' exists $\Rightarrow \exists c\in(a,b): f'(c)=0$

★ board favourite

Mean Value Theorem (Lagrange)

$f'(c) = \frac{f(b)-f(a)}{b-a}$

★ board favourite

Continuity at x = a

$\lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) = f(a)$

★ frequently asked

d/dx(cot⁻¹x)

$-\frac{1}{1+x^2}$

★ frequently asked

d/dx(sec⁻¹x)

$\frac{1}{x\sqrt{x^2-1}}$

★ frequently asked

d/dx(cosec⁻¹x)

$-\frac{1}{x\sqrt{x^2-1}}$

★ frequently asked

If y = (tan⁻¹x)², show that (1 + x²)² y₂ + 2x(1 + x²) y₁ = 2.

If x = a(cos θ + θ sin θ) and y = a(sin θ − θ cos θ), find d²y/dx².

If y = (sin x)^(cos x) + (cos x)^(sin x), find dy/dx using logarithmic differentiation.

Find the values of a and b such that the function f(x) = {5, if x ≤ 2; ax + b, if 2 < x < 10; 21, if x ≥ 10} is continuous.

Find the value of k so that f(x) = {kx+1, if x≤5; 3x−5, if x>5} is continuous at x=5.

If x^y + y^x = a^b, find dy/dx.

Verify Rolle's theorem for f(x) = x² − 4x + 3 on [1, 3].

Verify the Lagrange's Mean Value Theorem for the function f(x) = x² − 3x + 2 on the interval [−2, 3].

Show that the function f(x) = |x − 3| is continuous but not differentiable at x = 3.

If y = x^(cos x) + (cos x)^x, find dy/dx.

Differentiate sin(x²) with respect to x.

How many marks is Continuity and Differentiability worth in the Class 12 board exam?

Around 8 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Second derivative and Rolle's theorem proofs are frequent.

Is Continuity and Differentiability easy or hard?

It's rated hard — one of the tougher chapters in Class 12 Maths. Most students need about 5 hours to cover it well.

What are the important topics in Continuity and Differentiability?

The chapter covers 12 NCERT topics in 3 areas: Continuity; Differentiability and Standard Derivatives; Advanced Differentiation.

What questions come from Continuity and Differentiability in board exams?

Between 2018–2024, CBSE board papers asked questions from this chapter on Second Order Derivative, Parametric Differentiation, Differentiation, Continuity — 1- to 5-mark questions.