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Cheat sheet · JEE Preparation 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.

JEE · Mathematics

Cheat sheet

Class 12 · 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

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