Class 12 Maths · Chapter 5 Unit: Calculus Tentative Study Share: 7.0 Marks

Chapter 5: Continuity And Differentiability — NCERT Solutions, Notes, Formula Sheet & PYQs

Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Continuity And Differentiability.

📄 Printable Formula Sheet 🧠 Printable Mind Map ⚡ Practise in Interactive Hub →

NCERT Exercises in Chapter 5: Continuity And Differentiability

Exercise 5.1

34 NCERT Questions Solved

Exercise 5.2

10 NCERT Questions Solved

Exercise 5.3

15 NCERT Questions Solved

Exercise 5.4

10 NCERT Questions Solved

Exercise 5.5

18 NCERT Questions Solved

Exercise 5.6

11 NCERT Questions Solved

Exercise 5.7

17 NCERT Questions Solved

Miscellaneous Exercise

22 NCERT Questions Solved

📘 NCERT Concept Notes — Continuity And Differentiability

01 · Continuity joins value and limit

At a junction, compute the left limit, right limit and assigned value separately. A matching pair of limits is insufficient if the function value differs. Differentiability implies continuity; a continuous corner need not be differentiable.

02 · Differentiate layers, products and constraints

Name inner and outer functions before using the chain rule. For implicit equations, every derivative of a y-expression needs a factor y′. Keep denominator restrictions from the original function.

03 · Exponentials, logs and variable powers

Logarithmic differentiation is especially useful when both base and exponent vary. Use positive bases for general real powers. A formula obtained by dividing by the function must be rechecked at zeros.

04 · A parameter changes both coordinates

Differentiate x and y with respect to the same parameter. For the second derivative, differentiate the first slope with respect to the parameter, then divide by dx/dt again.

⚡ Key Formulas & Results — Continuity And Differentiability

Continuity at a point

Which three values must match at an interior junction?

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

Derivative from first principles

What limit defines the derivative?

\(f\prime(c)=\lim_{h\to0}\frac{f(c+h)-f(c)}h\)

Why differentiability implies continuity

How can the increment f(c+h)−f(c) be factored?

\(f(c+h)-f(c)=h\frac{f(c+h)-f(c)}h\longrightarrow0\)

Product, quotient and chain

Which factors must survive differentiation?

\((uv)\prime=u\prime v+uv\prime,\quad (u/v)\prime=\frac{u\prime v-uv\prime}{v^2},\quad(f(g))\prime=f\prime(g)g\prime\)

Inverse trigonometric derivatives

Recall the four inverse-trig derivatives used in this chapter.

\(\begin{aligned}(\sin^{-1}x)\prime&=(1-x^2)^{-1/2}\\(\cos^{-1}x)\prime&=-(1-x^2)^{-1/2}\\(\tan^{-1}x)\prime&=(1+x^2)^{-1}\\(\cot^{-1}x)\prime&=-(1+x^2)^{-1}\end{aligned}\)

Derive the sine-inverse derivative

Why is the square root positive in the denominator?

\(\frac{d}{dx}\sin^{-1}x=\frac1{\sqrt{1-x^2}}\)

Exponential and logarithmic derivatives

How do the bases affect these derivatives?

\((e^x)\prime=e^x,\quad(a^x)\prime=a^x\log a,\quad(\log x)\prime=1/x\)

Derive a variable power

How do you differentiate u(x) raised to v(x)?

\((u^v)\prime=u^v\left(v\prime\log u+v\frac{u\prime}{u}\right)\)

Parametric derivatives

How do you obtain the first and second x-derivatives?

\(\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\quad\frac{d^2y}{dx^2}=\frac{\frac d{dt}(dy/dx)}{dx/dt}\)