KTUYSBeta
Log inSign up

Cheat sheet · Class 11 · CBSE · Science Mathematics

Limits and Derivatives cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

CBSE Class 11 · Mathematics

Cheat sheet

Chapter 12Limits and Derivatives

Derivative = limit of (change in f) / (change in x) as the change shrinks to 0.

1. Standard Limits

lim⁡x→axn−anx−a=nan−1\lim_{x\to a}\dfrac{x^n - a^n}{x - a} = na^{n - 1}
lim⁡x→0sin⁡xx=1\lim_{x\to0}\dfrac{\sin x}{x} = 1lim⁡x→01−cos⁡xx=0\lim_{x\to0}\dfrac{1 - \cos x}{x} = 0

A limit exists when left- and right-hand limits are equal.

2. Derivative

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h\to0}\dfrac{f(x + h) - f(x)}{h}

3. Rules & Standard Derivatives

(uv)′=u′v+uv′(uv)' = u'v + uv'(uv)′=u′v−uv′v2\left(\dfrac uv\right)' = \dfrac{u'v - uv'}{v^2}
ddxxn=nxn−1\dfrac{d}{dx}x^n = nx^{n - 1}ddxsin⁡x=cos⁡x\dfrac{d}{dx}\sin x = \cos xddxcos⁡x=−sin⁡x\dfrac{d}{dx}\cos x = -\sin x
ddxtan⁡x=sec⁡2x\dfrac{d}{dx}\tan x = \sec^2xddxsec⁡x=sec⁡xtan⁡x\dfrac{d}{dx}\sec x = \sec x\tan x

4. Key Points

  • Derivative of a constant is 0.
  • Algebra of limits: sum, product, quotient.
  • Rationalise for 0/0 with roots.
  • Derivative = slope of the tangent.

Remember: sin x/x → 1 | first principle | product & quotient rules

Limits and Derivatives: questions and notes

More Class 11 · CBSE · Science Mathematics cheat sheets

Practise Limits and Derivatives

Sign up free to practise the full chapter. Questions you get wrong come back until you've mastered them.

Start practising free

Free to start · No credit card needed