Knowledge organisers

Continuity and Differentiability: CBSE Class 12 Maths knowledge organiser

Everything to know about continuity and differentiability 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

Continuous function: f is continuous at c when the limit of f(x) as x → c equals f(c).
Differentiable: The derivative exists at the point; differentiable implies continuous.
Chain rule: d/dx f(g(x)) = f′(g(x)) g′(x).

Key formulas

Continuous at \(c\)	\(\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c)\)
Differentiable at \(c\): LHD = RHD; differentiable ⇒ continuous (not conversely, e.g. \(\|x\|\) at 0)
Standard	\((x^n)'=nx^{n-1},\) \((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((\tan x)'=\sec^2x\)
More trig	\((\cot x)'=-\cosec^2x,\) \((\sec x)'=\sec x\tan x,\) \((\cosec x)'=-\cosec x\cot x\)
Exp, log	\((e^x)'=e^x,\) \((a^x)'=a^x\log a,\) \((\log x)'=\tfrac1x,\) \((\log_ax)'=\tfrac1{x\log a}\)
Inverse trig	\((\sin^{-1}x)'=\tfrac1{\sqrt{1-x^2}},\) \((\cos^{-1}x)'=-\tfrac1{\sqrt{1-x^2}},\) \((\tan^{-1}x)'=\tfrac1{1+x^2}\)
Rules	\((uv)'=u'v+uv',\) \(\Big(\frac uv\Big)'=\frac{u'v-uv'}{v^2},\) \(\frac{dy}{dx}=\frac{dy}{dt}\cdot\frac{dt}{dx}\)
Logarithmic, \(y=u^v\)	\(\log y=v\log u\Rightarrow\frac1y\frac{dy}{dx}=v'\log u+\frac{vu'}u\)
Parametric	\(\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\) \(\frac{d^2y}{dx^2}=\frac{\frac d{dt}\left(\frac{dy}{dx}\right)}{dx/dt}\)
Implicit: differentiate term by term, \(\frac{d}{dx}(y^2)=2y\frac{dy}{dx}\)

Worked example

Is \(f(x) = \begin{cases} 3x - 2, & x \le 2 \\ x^2, & x > 2\end{cases}\) differentiable at \(2\)? \(f(2) = 4\), right limit \(4\): continuous. LHD \(= 3\), RHD \(= 4\): not differentiable.

Common mistakes

Declaring a function differentiable because the two formulas "join up" — continuity is not enough.
Using \(\log(u+v) = \log u + \log v\) (false). Take logs only of products, quotients and powers.
Treating \(x^x\) like \(x^n\) or \(a^x\): neither rule applies; use logarithms.
Forgetting \(\frac{dy}{dx}\) on \(y\)-terms in implicit differentiation, or the inner derivative in the chain rule.

You should be able to…

Check continuity at a point, find k for continuity and differentiate with the chain rule, logs and exponentials.
Test differentiability, and use implicit, logarithmic, parametric and inverse-trig differentiation on board-standard questions.
Write long answers and case studies in full, especially 'prove that' second-derivative questions and xˣ-type derivatives.
Handle piecewise functions with two unknowns, |x| and greatest-integer functions and max/min-defined functions.

The printable sheet

Continuity and Differentiability knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Continuity and Differentiability knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers