Class 12 Maths Notes

Continuity and Differentiability Class 12 Notes

Complete, exam-ready notes on continuity and differentiability: the limit definition of continuity, differentiability and its link to continuity, the standard differentiation rules, and logarithmic, implicit and parametric differentiation — written for CBSE boards and JEE revision.

Class12SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

Continuity and Differentiability studies when a function's graph has no breaks (continuous) and when it has a well-defined slope (differentiable) — the foundation of all calculus in Class 12.

Continuity of a Function

Continuity at a point

\lim_{x\to a}f(x) = f(a)

A function is continuous at x = a when it is defined at a, the limit exists at a, and the two are equal. All polynomial, rational, trigonometric, exponential and logarithmic functions are continuous on their domains.

Left-hand limit and right-hand limit must both equal f(a).
Sums, differences, products and quotients (with non-zero denominator) of continuous functions are continuous.
A common composite rule:
$\lim_{x\to a} f(g(x)) = f\left(\lim_{x\to a} g(x)\right)$
when f is continuous.

Classic trap: |x| at 0

The absolute value function is continuous at x = 0 but NOT differentiable there — the left and right slopes differ. Continuity does not guarantee differentiability.

Differentiability

Derivative at a point

f'(a) = \lim_{h\to 0}\frac{f(a+h) - f(a)}{h}

The derivative is the slope of the tangent, or the limit of the secant slope. Differentiability at a point implies continuity there, but the converse is false.

A function is differentiable on an open interval if it is differentiable at every point of it.
If the left and right derivatives differ, the function is not differentiable (a sharp corner).
Standard derivatives:
$\frac{d}{dx}x^n = nx^{n-1}$
,
$\frac{d}{dx}\sin x = \cos x$
,
$\frac{d}{dx}e^x = e^x$
.

f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}

Definition of the derivative

Rules of Differentiation

Chain rule:
$\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$
.
Product rule:
$(uv)' = u'v + uv'$
.
Quotient rule:
$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
.
Derivatives of inverse trig functions and the special forms
$\frac{d}{dx}\ln x = \frac{1}{x}$
,
$\frac{d}{dx}a^x = a^x\ln a$
.

\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx},\quad (uv)' = u'v + uv'

Chain and product rules

Logarithmic, Implicit and Parametric Differentiation

Logarithmic differentiation: take ln of both sides first, then differentiate — ideal for
$x^x$
or long products and powers.
Implicit differentiation: differentiate y(x) terms using the chain rule, then solve for
$dy/dx$
.
Parametric form: if
$x = f(t)$
,
$y = g(t)$
then
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
.
The second derivative in parametric form:
$\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy}{dx}\right)\Big/\frac{dx}{dt}$
(with care about the chain rule).

\frac{dy}{dx} = \frac{dy/dt}{dx/dt},\quad \frac{d}{dx}(\ln y) = \frac{1}{y}\frac{dy}{dx}

Parametric and logarithmic differentiation

Solved Examples

Example: Find $\frac{dy}{dx}$ if $y = \sin(x^2 + 1)$ .

Solution: Using the chain rule with $u = x^2 + 1$ : $\frac{dy}{dx} = \cos(x^2+1) \cdot 2x = 2x\cos(x^2+1)$ .

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Continuity

\lim_{x\to a}f(x) = f(a)

Derivative definition

f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}

Chain rule

\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}

Product rule

(uv)' = u'v + uv'

Quotient rule

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Parametric form

\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

Power rule

\frac{d}{dx}x^n = nx^{n-1}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

Differentiability implies continuity, but not vice versa.
A sharp corner (like |x| at 0) is continuous but not differentiable.
Check continuity via LHL and RHL separately.
Use the chain rule whenever a function is composed, even implicitly.
Logarithmic differentiation handles x^x and products of many factors.
Parametric slope = (dy/dt)/(dx/dt).

FAQ

Common questions

Does continuity imply differentiability?

No. Continuity at a point only guarantees the limit equals the function value. A function like |x| is continuous but not differentiable at x = 0 because the left and right slopes differ.

How do I check if a function is continuous at a point?

Verify the function is defined there, the left-hand and right-hand limits both exist and agree, and their common value equals f(a).

When is logarithmic differentiation used?

For expressions like x^x, or a product of many factors, taking ln on both sides converts products into sums before differentiating — much simpler.

What does parametric differentiation mean?

When x and y are both given in terms of a parameter t, the slope is dy/dx = (dy/dt)/(dx/dt) rather than a direct derivative of y with respect to x.

Continue learning

Mastering this chapter with live help

Notes help, but doubts clear fastest in a live class. Narayan Gurukul Academy (ClassApna) runs small-batch CBSE, JEE and NEET coaching from our Mohali centre and online — with daily doubt support and mock tests.

Book a Free Demo See plans & pricing

One-on-one guidance available · Live online classes across India