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Cheat sheet · JEE Preparation 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.

JEE · Mathematics

Cheat sheet

Class 11 · 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

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