KTUYSBeta
Log inSign up

Cheat sheet · Class 12 · CBSE · Science Mathematics

Continuity and Differentiability cheat sheet

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

CBSE Class 12 · Mathematics

Cheat sheet

Chapter 5Continuity and Differentiability

Differentiable ⇒ continuous, but not the other way (|x| at 0).

1. Continuity

ff is continuous at aa if lim⁡x→a−f=lim⁡x→a+f=f(a)\lim_{x\to a^-}f = \lim_{x\to a^+}f = f(a).

Sums, products and quotients of continuous functions are continuous.

2. Derivatives

FunctionDerivative
sin⁡−1x\sin^{-1}x11−x2\dfrac{1}{\sqrt{1 - x^2}}
cos⁡−1x\cos^{-1}x−11−x2-\dfrac{1}{\sqrt{1 - x^2}}
tan⁡−1x\tan^{-1}x11+x2\dfrac{1}{1 + x^2}
exe^xexe^x
log⁡x\log x1x\dfrac1x

3. Techniques

  • Chain rule: dydx=dydu⋅dudx\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}.
  • Implicit differentiation.
  • Logarithmic: for uvu^v take logs first.
  • Parametric: dydx=dy/dtdx/dt\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}.
  • Second derivative: differentiate again.

4. Key Points

  • |x| is continuous but not differentiable at 0.
  • Polynomials are continuous everywhere.
  • eˣ and log x are continuous on their domains.
  • Logs simplify products and powers.

Remember: Continuity test | inverse-trig derivatives | chain, log, parametric

Continuity and Differentiability: questions and notes

More Class 12 · CBSE · Science Mathematics cheat sheets

Practise Continuity and Differentiability

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