Knowledge organisers

Limits and Derivatives: CBSE Class 11 Maths knowledge organiser

Everything to know about limits and derivatives on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Limit: The value f(x) approaches as x approaches a.
Derivative: The rate of change of f at a point: the limit of (f(x + h) − f(x))/h as h → 0.
First principles: Finding a derivative directly from the limit definition.

Key formulas

Standard limits	\(\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1},\) \(\lim_{x\to0}\frac{\sin x}x=1\)
Definition	\(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h\)
Rules	\((uv)'=u'v+uv',\) \(\left(\frac uv\right)'=\frac{u'v-uv'}{v^2}\)
Derivatives	\((x^n)'=nx^{n-1},\ (\sin x)'=\cos x,\ (\cos x)'=-\sin x,\ (\tan x)'=\sec^2x\)

Worked example

Find \(\displaystyle\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\).

\(\dfrac{(x - 3)(x + 3)}{x - 3} = x + 3 \to 6\).

Common mistakes

Writing \(\dfrac00 = 0\) or \(1\): it means "simplify first".
\(\dfrac{\sin 3x}{x} \to 1\): it tends to \(3\).
Product rule applied as \((uv)' = u'v'\); quotient rule with the numerator the wrong way round.
\(\dfrac{d}{dx}\cos x = \sin x\): the sign is negative.

You should be able to…

Evaluate standard limits by substitution or factorising and differentiate polynomials and sin, cos.
Use sin x/x → 1, differentiate from first principles and apply the product and quotient rules.
Handle limits needing surds or trigonometric rewriting and use derivatives as rates of change.
Find constants that make a limit exist and prove standard derivatives from the definition.

The printable sheet

Limits and Derivatives knowledge organiser for CBSE Class 11 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Limits and Derivatives knowledge organiser (CBSE Class 11 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 11 Maths topics: Sets · Relations and Functions · Trigonometric Functions · Complex Numbers and Quadratic Equations · Linear Inequalities · Permutations and Combinations · Binomial Theorem · Sequences and Series · Straight Lines · Conic Sections · Introduction to Three Dimensional Geometry · Statistics · Probability · All CBSE Class 11 Maths organisers