Mathematics · JEE

Limit, Continuity and Differentiability Formula Sheet for JEE

68+ JEE formulas in this unit

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The Limit, Continuity and Differentiability JEE formula sheet lists 68+ important formulas for JEE Main and Advanced, including essential identities from Real-valued functions, algebra of functions, Polynomial, rational, trigonometric, logarithmic and exponential functions, Inverse functions, Graphs of simple functions, and more. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Limit, Continuity and Differentiability. This unit-wise formula list covers 68+ exam-relevant results across Real-valued functions, algebra of functions, Polynomial, rational, trigonometric, logarithmic and exponential functions, Inverse functions, Graphs of simple functions, and more, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

68 formulas across 9 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 43Important: 19Supplementary: 6

Real-valued functions, algebra of functions

Domain of algebraic combination of functionsEssential

\begin{aligned}D(f\pm g)&=D(f)\cap D(g),\\ D(fg)&=D(f)\cap D(g),\\ D\!\left(\frac{f}{g}\right)&=\{x\in D(f)\cap D(g):g(x)\neq 0\}\end{aligned}

Variables: $D(f)$ denotes domain of function $f$ .
Conditions: Applicable for real-valued functions where the operations are defined.
Where used in JEE: Finding domain of combined functions and checking validity before limits/continuity/differentiation.

Composition of functions and domainEssential

D(f\circ g)=\{x\in D(g):g(x)\in D(f)\},\qquad (f\circ g)(x)=f(g(x))

Variables: $f\circ g$ is composition of $f$ and $g$ .
Conditions: Defined only when output of $g$ lies in domain of $f$ .
Where used in JEE: Composite functions, chain rule, inverse functions, domain/range questions.

Even and odd functionsImportant

f\text{ even}\iff f(-x)=f(x),\qquad f\text{ odd}\iff f(-x)=-f(x)

Variables: $x$ belongs to a domain symmetric about origin.
Conditions: Domain should be symmetric with respect to $0$ .
Where used in JEE: Graph symmetry, simplification of expressions, continuity and differentiability checks.

Periodicity of a functionImportant

f(x+T)=f(x)\quad \forall x\text{ in domain, with }T>0

Variables: $T$ is a period; least positive period, if it exists, is fundamental period.
Conditions: Must hold for all admissible $x$ .
Where used in JEE: Trigonometric functions, graph sketching, limits and derivatives of periodic functions.

Identity function, modulus and signumImportant

I(x)=x,\qquad |x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases},\qquad \operatorname{sgn}(x)=\begin{cases}1,&x>0\\0,&x=0\\-1,&x<0\end{cases}

Variables: $|x|$ is modulus and $\operatorname{sgn}(x)$ is signum.
Where used in JEE: Piecewise continuity/differentiability, graph-based questions, derivative of modulus forms.

Greatest integer and fractional partImportant

x=[x]+\{x\},\qquad [x]\le x<[x]+1,\qquad 0\le \{x\}<1

Variables: $[x]$ is greatest integer function, $\{x\}$ is fractional part.
Conditions: For real $x$ .
Where used in JEE: Discontinuity and graph questions, limits involving piecewise jumps.

Polynomial, rational, trigonometric, logarithmic and exponential functions

Polynomial function factsEssential

p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0

Variables: $a_n\neq 0$ , degree $n$ .
Conditions: Defined for all real $x$ .
Where used in JEE: Continuity, differentiability, graph shape, end behavior and derivative computations.

Rational function and domainEssential

r(x)=\frac{p(x)}{q(x)},\qquad D(r)=\{x:q(x)\neq 0\}

Variables: $p,q$ are polynomials.
Conditions: Exclude zeros of denominator.
Where used in JEE: Domain, continuity, discontinuity, asymptotes, differentiability.

Basic trigonometric functions and domainsEssential

\sin x,\cos x\text{ defined }\forall x;\quad \tan x=\frac{\sin x}{\cos x},\ \sec x=\frac1{\cos x},\ \csc x=\frac1{\sin x},\ \cot x=\frac{\cos x}{\sin x}

Variables: $x$ in radians for calculus formulas.
Conditions: $\tan x,\sec x$ undefined when $\cos x=0$ ; $\csc x,\cot x$ undefined when $\sin x=0$ .
Where used in JEE: Limits, continuity, derivatives and graph questions.

Fundamental periods of trigonometric functionsImportant

T_{\sin x}=2\pi,\quad T_{\cos x}=2\pi,\quad T_{\tan x}=\pi,\quad T_{\cot x}=\pi,\quad T_{\sec x}=2\pi,\quad T_{\csc x}=2\pi

Variables: $T$ denotes fundamental period.
Where used in JEE: Graphs, simplification, periodic limit/derivative questions.

Exponential function basicsEssential

a^x=e^{x\log a}\ (a>0,a\neq 1),\qquad e^x>0\ \forall x

Variables: $a$ is positive base, $e$ is Euler's number.
Conditions: Real-valued exponential requires $a>0$ .
Where used in JEE: Transforming exponentials, limits, differentiation, monotonicity.

Logarithmic function basicsEssential

y=\log_a x=\frac{\ln x}{\ln a}\ (a>0,a\neq 1),\qquad \ln x=\log_e x

Variables: $a$ is base, $x>0$ .
Conditions: Logarithm defined only for positive argument.
Where used in JEE: Change of base, derivatives, continuity and inverse function questions.

Logarithm lawsImportant

\log_a(xy)=\log_a x+\log_a y,\quad \log_a\!\left(\frac{x}{y}\right)=\log_a x-\log_a y,\quad \log_a(x^r)=r\log_a x

Variables: $a>0,a\neq 1$ , $x,y>0$ , $r\in\mathbb R$ .
Conditions: Arguments must be positive in real domain.
Where used in JEE: Log differentiation, simplification before limits and derivatives.

Inverse functions

Inverse function criterionEssential

f^{-1}(f(x))=x\ (x\in D(f)),\qquad f(f^{-1}(y))=y\ (y\in R(f))

Variables: $D(f)$ and $R(f)$ denote domain and range.
Conditions: Inverse exists when $f$ is one-one and onto its range.
Where used in JEE: Finding inverse, composition, derivative of inverse, solving equations.

Derivative of inverse functionImportant

\left(f^{-1}\right)'(y)=\frac{1}{f'(x)}\quad \text{where }y=f(x)

Variables: $y=f(x)$ , $f^{-1}$ is inverse of $f$ .
Conditions: $f$ must be invertible and differentiable at $x$ with $f'(x)\neq 0$ .
Where used in JEE: Derivative of inverse functions, inverse trigonometric derivative derivations and applications.

Inverse trigonometric principal domains and rangesEssential

\begin{aligned}&y=\sin^{-1}x\iff x=\sin y,\ y\in\left[-\frac\pi2,\frac\pi2\right],\ x\in[-1,1]\\&y=\cos^{-1}x\iff x=\cos y,\ y\in[0,\pi],\ x\in[-1,1]\\&y=\tan^{-1}x\iff x=\tan y,\ y\in\left(-\frac\pi2,\frac\pi2\right),\ x\in\mathbb R\\&y=\cot^{-1}x\iff x=\cot y,\ y\in(0,\pi),\ x\in\mathbb R\\&y=\sec^{-1}x\iff x=\sec y,\ y\in[0,\pi]\setminus\left\{\frac\pi2\right\},\ |x|\ge 1\\&y=\csc^{-1}x\iff x=\csc y,\ y\in\left[-\frac\pi2,\frac\pi2\right]\setminus\{0\},\ |x|\ge 1\end{aligned}

Variables: Principal values are chosen to make each inverse single-valued.
Conditions: Use principal branch conventions.
Where used in JEE: Inverse trig equations, graph, differentiation and continuity questions.

Standard identities of inverse trigonometric functionsImportant

\sin^{-1}x+\cos^{-1}x=\frac\pi2,\qquad \tan^{-1}x+\cot^{-1}x=\frac\pi2

Variables: $x$ real where expressions are defined.
Conditions: First identity for $x\in[-1,1]$ ; second for all real $x$ under principal values.
Where used in JEE: Simplification in limits and derivatives.

Sum identities for inverse tangentSupplementary

\tan^{-1}x+\tan^{-1}y=\tan^{-1}\!\left(\frac{x+y}{1-xy}\right)

Variables: $x,y\in\mathbb R$ .
Conditions: Principal value adjustment by adding/subtracting $\pi$ may be needed depending on signs and quadrant; if $xy<1$ and principal value lies in range, direct use is valid.
Where used in JEE: Simplifying inverse tangent expressions in limits and identities.

Graphs of simple functions

Basic graph transformationsEssential

\begin{aligned}y=f(x-a)&\Rightarrow \text{shift right by }a,\\ y=f(x)+b&\Rightarrow \text{shift up by }b,\\ y=-f(x)&\Rightarrow \text{reflection in }x\text{-axis},\\ y=f(-x)&\Rightarrow \text{reflection in }y\text{-axis},\\ y=|f(x)|&\Rightarrow \text{portion below }x\text{-axis reflected above},\\ y=f(|x|)&\Rightarrow \text{right half mirrored to left}\end{aligned}

Variables: $a,b\in\mathbb R$ .
Where used in JEE: Sketching simple graphs and identifying continuity/differentiability points.

Graphs of standard functionsImportant

y=x,\ y=x^2,\ y=x^3,\ y=|x|,\ y=[x],\ y=\{x\},\ y=\frac1x,\ y=\sqrt{x},\ y=e^x,\ y=\ln x,\ y=\sin x,\ y=\cos x,\ y=\tan x

Variables: Standard elementary functions.
Conditions: Use usual domains of the listed functions.
Where used in JEE: Graph-based continuity, domain-range, monotonicity and differentiability problems.

Continuity and differentiability of greatest integer functionImportant

[x]\text{ is continuous on }\mathbb R\setminus\mathbb Z,\text{ discontinuous at every integer, and }\frac{d}{dx}[x]=0\text{ for }x\notin\mathbb Z

Variables: $\mathbb Z$ is set of integers.
Conditions: Derivative fails at integers due to discontinuity.
Where used in JEE: Standard graph, continuity and differentiability questions.

Differentiability of fractional part functionSupplementary

\{x\}=x-[x],\quad \{x\}\text{ is continuous on }\mathbb R\setminus\mathbb Z,\text{ discontinuous at integers, and }\frac{d}{dx}\{x\}=1\text{ for }x\notin\mathbb Z

Variables: $\{x\}$ is fractional part.
Conditions: Derivative not defined at integers.
Where used in JEE: Discontinuity and derivative of piecewise periodic functions.

Limits, continuity and differentiability

Definition of limitEssential

\lim_{x\to a}f(x)=L

Variables: $L$ is the value approached by $f(x)$ as $x\to a$ .
Conditions: The limit may exist even if $f(a)$ is undefined or different from $L$ .
Where used in JEE: Core concept for continuity and derivative definition.

One-sided limits and existence of limitEssential

\lim_{x\to a}f(x)\text{ exists }\iff \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)

Variables: Left-hand limit $x\to a^-$ , right-hand limit $x\to a^+$ .
Conditions: Both one-sided limits must be finite and equal for finite two-sided limit.
Where used in JEE: Piecewise functions and continuity/discontinuity at junction points.

Algebra of limitsEssential

\begin{aligned}\lim(f\pm g)&=\lim f\pm \lim g,\\ \lim(cf)&=c\lim f,\\ \lim(fg)&=(\lim f)(\lim g),\\ \lim\left(\frac{f}{g}\right)&=\frac{\lim f}{\lim g}\quad (\lim g\neq 0)\end{aligned}

Variables: Limits are taken as $x\to a$ .
Conditions: Assuming the individual limits exist and denominator limit is nonzero for quotient.
Where used in JEE: Direct evaluation of limits.

Limit of polynomial, rational and composite functionsEssential

\lim_{x\to a}p(x)=p(a),\qquad \lim_{x\to a}\frac{p(x)}{q(x)}=\frac{p(a)}{q(a)}\ (q(a)\neq 0),\qquad \lim_{x\to a}f(g(x))=f\!\left(\lim_{x\to a}g(x)\right)

Variables: $p,q$ polynomials.
Conditions: For composition, $f$ should be continuous at $\lim g(x)$ .
Where used in JEE: Evaluating standard limits quickly.

Standard trigonometric limitsEssential

\lim_{x\to 0}\frac{\sin x}{x}=1,\qquad \lim_{x\to 0}\frac{\tan x}{x}=1,\qquad \lim_{x\to 0}\frac{1-\cos x}{x}=0,\qquad \lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac12

Variables: $x$ measured in radians.
Conditions: Radian measure is essential.
Where used in JEE: Most standard JEE limit and derivative derivations.

Derived small-angle limitsEssential

\lim_{x\to 0}\frac{\sin ax}{ax}=1,\quad \lim_{x\to 0}\frac{\sin ax}{x}=a,\quad \lim_{x\to 0}\frac{\tan ax}{x}=a,\quad \lim_{x\to 0}\frac{1-\cos ax}{x^2}=\frac{a^2}{2}

Variables: $a\in\mathbb R$ constant.
Conditions: Radian measure.
Where used in JEE: Direct substitution after standard trigonometric limits.

Exponential and logarithmic standard limitsEssential

\lim_{x\to 0}\frac{e^x-1}{x}=1,\qquad \lim_{x\to 0}\frac{a^x-1}{x}=\ln a,\qquad \lim_{x\to 0}\frac{\ln(1+x)}{x}=1

Variables: $a>0,a\neq 1$ .
Conditions: For real logarithm, $1+x>0$ near $0$ .
Where used in JEE: Limits, derivatives of exponential/logarithmic functions.

Binomial-type and power limitsImportant

\lim_{x\to 0}\frac{(1+x)^n-1}{x}=n,\qquad \lim_{x\to 0}\frac{(1+x)^\alpha-1}{x}=\alpha

Variables: $n\in\mathbb N$ , $\alpha\in\mathbb R$ .
Conditions: For real $\alpha$ , require $1+x>0$ near $0$ .
Where used in JEE: Simplification of algebraic and exponential limits.

Fundamental exponential limitsEssential

\lim_{x\to \infty}\left(1+\frac1x\right)^x=e,\qquad \lim_{x\to 0}(1+x)^{1/x}=e

Variables: $e$ is Euler's number.
Conditions: Second limit uses $1+x>0$ near $0$ .
Where used in JEE: Evaluating indeterminate forms of type $1^\infty$.

Generalized exponential limitImportant

\lim_{x\to \infty}\left(1+\frac{a}{x}\right)^x=e^a,\qquad \lim_{x\to 0}(1+ax)^{1/x}=e^a

Variables: $a\in\mathbb R$ .
Conditions: For second form, $1+ax>0$ near the limit point.
Where used in JEE: Indeterminate forms and sequence/function limits.

Continuity at a pointEssential

f\text{ is continuous at }x=a\iff \lim_{x\to a}f(x)=f(a)

Variables: $a$ is a point in domain of $f$ .
Conditions: Equivalent to existence of $f(a)$ and equality of left and right limits with it.
Where used in JEE: Checking continuity of elementary and piecewise functions.

Continuity of elementary functionsEssential

\text{Polynomials, exponential, sine, cosine are continuous on their domains; rational, logarithmic, tangent, cotangent, secant, cosecant are continuous wherever defined.}

Conditions: Exclude points outside domain or denominator-zero points.
Where used in JEE: Direct continuity conclusions without limit computation.

Differentiability implies continuityEssential

f\text{ differentiable at }a\implies f\text{ continuous at }a

Variables: $a$ is a point in domain.
Conditions: Converse is not always true.
Where used in JEE: Theory-based MCQs and checking non-differentiability quickly.

Derivative from first principleEssential

f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}

Variables: $h$ is increment in independent variable.
Conditions: Limit must exist finitely.
Where used in JEE: Definition-based derivative problems and derivation of basic formulas.

Left and right derivativesEssential

f'_{-}(a)=\lim_{h\to 0^-}\frac{f(a+h)-f(a)}{h},\qquad f'_{+}(a)=\lim_{h\to 0^+}\frac{f(a+h)-f(a)}{h}

Variables: $f'_{-}(a)$ and $f'_{+}(a)$ are left and right derivatives.
Conditions: $f$ differentiable at $a$ iff both exist and are equal.
Where used in JEE: Piecewise functions, modulus and corner/cusp points.

Differentiability of modulus compositionImportant

\frac{d}{dx}|f(x)|=\frac{f(x)}{|f(x)|}f'(x)=\operatorname{sgn}(f(x))f'(x)\quad (f(x)\neq 0)

Variables: $f$ differentiable function.
Conditions: At points where $f(x)=0$ , differentiability must be checked separately.
Where used in JEE: Piecewise derivative, cusp/corner analysis and chain-rule problems.

Differentiation of the sum, difference, product and quotient of two functions

Derivative of constant and identityEssential

\frac{d}{dx}(c)=0,\qquad \frac{d}{dx}(x)=1

Variables: $c$ constant.
Where used in JEE: Basic differentiation.

Linearity of differentiationEssential

\frac{d}{dx}(af(x)\pm bg(x))=a f'(x)\pm b g'(x)

Variables: $a,b$ constants.
Conditions: $f,g$ differentiable.
Where used in JEE: Differentiating linear combinations of functions.

Product ruleEssential

\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}

Variables: $u=u(x),\ v=v(x)$ .
Conditions: $u,v$ differentiable.
Where used in JEE: Products of algebraic, trigonometric, exponential and logarithmic functions.

Quotient ruleEssential

\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\,u'-u\,v'}{v^2}

Variables: $u=u(x),\ v=v(x)$ .
Conditions: $u,v$ differentiable and $v\neq 0$ .
Where used in JEE: Differentiating rational and ratio-type functions.

Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions

Power ruleEssential

\frac{d}{dx}(x^n)=nx^{n-1}

Variables: $n\in\mathbb R$ in standard use.
Conditions: For non-integer real $n$ , require domain where $x^n$ is real-valued and differentiable.
Where used in JEE: Algebraic differentiation, composite functions, higher derivatives.

Derivatives of basic trigonometric functionsEssential

\frac{d}{dx}(\sin x)=\cos x,\quad \frac{d}{dx}(\cos x)=-\sin x,\quad \frac{d}{dx}(\tan x)=\sec^2x,\quad \frac{d}{dx}(\cot x)=-\csc^2x

Variables: $x$ in radians.
Conditions: At points where the functions are defined.
Where used in JEE: Routine differentiation and applications.

Derivatives of secant and cosecantEssential

\frac{d}{dx}(\sec x)=\sec x\tan x,\qquad \frac{d}{dx}(\csc x)=-\csc x\cot x

Variables: $x$ in radians.
Conditions: At points where the functions are defined.
Where used in JEE: Trigonometric differentiation.

Derivatives of inverse trigonometric functionsEssential

\begin{aligned}\frac{d}{dx}(\sin^{-1}x)&=\frac1{\sqrt{1-x^2}},\\ \frac{d}{dx}(\cos^{-1}x)&=-\frac1{\sqrt{1-x^2}},\\ \frac{d}{dx}(\tan^{-1}x)&=\frac1{1+x^2},\\ \frac{d}{dx}(\cot^{-1}x)&=-\frac1{1+x^2},\\ \frac{d}{dx}(\sec^{-1}x)&=\frac1{|x|\sqrt{x^2-1}},\\ \frac{d}{dx}(\csc^{-1}x)&=-\frac1{|x|\sqrt{x^2-1}}\end{aligned}

Variables: $x$ is real in the domain of each inverse function.
Conditions: For $\sin^{-1}x,\cos^{-1}x$ : $|x|<1$ ; for $\sec^{-1}x,\csc^{-1}x$ : $|x|>1$ .
Where used in JEE: Standard derivative evaluation and chain rule problems.

Derivatives of logarithmic functionsEssential

\frac{d}{dx}(\ln x)=\frac1x,\qquad \frac{d}{dx}(\log_a x)=\frac{1}{x\ln a}

Variables: $a>0,a\neq 1$ .
Conditions: Require $x>0$ .
Where used in JEE: Logarithmic differentiation and direct differentiation.

Derivatives of exponential functionsEssential

\frac{d}{dx}(e^x)=e^x,\qquad \frac{d}{dx}(a^x)=a^x\ln a

Variables: $a>0,a\neq 1$ .
Where used in JEE: Growth-decay type functions, chain rule applications.

Chain ruleEssential

\frac{d}{dx}f(g(x))=f'(g(x))\,g'(x)

Variables: $f$ outer function, $g$ inner function.
Conditions: Both relevant derivatives must exist.
Where used in JEE: Composite functions in all differentiation questions.

Generalized chain-rule formsImportant

\frac{d}{dx}[u(x)]^n=n[u(x)]^{n-1}u'(x),\qquad \frac{d}{dx}\sin u=\cos u\,u',\qquad \frac{d}{dx}e^{u}=e^{u}u'

Variables: $u=u(x)$ , $n\in\mathbb R$ .
Conditions: Where expressions are defined and differentiable.
Where used in JEE: Quick application of chain rule.

Derivative of logarithm of a functionEssential

\frac{d}{dx}\ln|u|=\frac{u'}{u}

Variables: $u=u(x)$ .
Conditions: $u\neq 0$ .
Where used in JEE: Logarithmic differentiation, derivatives of products/quotients/powers.

Logarithmic differentiation for variable powersImportant

y=[u(x)]^{v(x)}\Rightarrow \ln y=v\ln u\Rightarrow \frac{y'}{y}=v'\ln u+v\frac{u'}{u}

Variables: $u=u(x)>0,\ v=v(x)$ .
Conditions: For real-valued $u^v$ , typically take $u>0$ .
Where used in JEE: Functions of the type $f(x)^{g(x)}$, products of many factors, powers with variable exponents.

Implicit differentiationEssential

F(x,y)=0\Rightarrow \frac{dy}{dx}=-\frac{F_x}{F_y}

Variables: $F_x=\partial F/\partial x$ , $F_y=\partial F/\partial y$ .
Conditions: $F_y\neq 0$ and relation defines $y$ locally as differentiable function of $x$ .
Where used in JEE: Curves not given explicitly, tangent/normal questions.

Parametric differentiationSupplementary

\frac{dy}{dx}=\frac{dy/dt}{dx/dt}

Variables: $x=x(t),\ y=y(t)$ .
Conditions: $dx/dt\neq 0$ .
Where used in JEE: Parametric curves and related rate of change problems.

Derivatives of order upto two

Second derivative definitionEssential

\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=y''

Variables: $y=y(x)$ .
Conditions: Requires first derivative differentiable.
Where used in JEE: Concavity, maxima-minima test, curve analysis.

Second derivative under parametric formSupplementary

\frac{d^2y}{dx^2}=\frac{\dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)}{dx/dt}

Variables: $x=x(t),\ y=y(t)$ .
Conditions: $dx/dt\neq 0$ .
Where used in JEE: Parametric curvature/concavity and tangent questions.

Second derivative by implicit differentiationSupplementary

\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)

Variables: $y$ is implicitly defined by a relation in $x,y$ .
Conditions: Differentiate the obtained $dy/dx$ again, using implicit differentiation where needed.
Where used in JEE: Higher derivatives of implicit curves.

Successive derivatives of standard functionsImportant

\frac{d^2}{dx^2}(\sin x)=-\sin x,\quad \frac{d^2}{dx^2}(\cos x)=-\cos x,\quad \frac{d^2}{dx^2}(e^x)=e^x,\quad \frac{d^2}{dx^2}(\ln x)=-\frac1{x^2}

Conditions: At points where defined; for $\ln x$ , $x>0$ .
Where used in JEE: Direct second derivative evaluation.

Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable

Rate of change interpretationEssential

\text{Average rate of change on }[a,b]=\frac{f(b)-f(a)}{b-a},\qquad \text{Instantaneous rate of change at }x=a=f'(a)

Variables: $f$ is a function of one variable.
Conditions: For instantaneous rate, derivative should exist at $a$ .
Where used in JEE: Physical interpretation and application problems.

Related rates chain relationEssential

\frac{dy}{dt}=\frac{dy}{dx}\cdot\frac{dx}{dt}

Variables: $x,y$ depend on parameter/time $t$ .
Conditions: Relevant derivatives exist.
Where used in JEE: Rate of change of connected quantities.

Increasing and decreasing via first derivativeEssential

f'(x)>0\Rightarrow f\text{ increasing},\qquad f'(x)<0\Rightarrow f\text{ decreasing}

Conditions: On an interval where derivative sign is maintained.
Where used in JEE: Monotonicity and interval analysis.

Non-decreasing and non-increasing criteriaImportant

f'(x)\ge 0\Rightarrow f\text{ non-decreasing},\qquad f'(x)\le 0\Rightarrow f\text{ non-increasing}

Conditions: On an interval where derivative exists and satisfies the inequality.
Where used in JEE: Theoretical monotonicity questions.

Critical pointsEssential

\text{Critical points occur where }f'(x)=0\text{ or }f'(x)\text{ does not exist}

Conditions: Point should belong to domain of $f$ .
Where used in JEE: Finding maxima, minima and monotonic intervals.

First derivative test for local extremaEssential

\begin{aligned}f'\text{ changes }+\to -&\Rightarrow \text{local maximum},\\ f'\text{ changes }-\to +&\Rightarrow \text{local minimum},\\ \text{no sign change}&\Rightarrow \text{no extremum at that critical point}\end{aligned}

Variables: Signs are checked around a critical point.
Conditions: Requires sign analysis of $f'$ in neighborhoods.
Where used in JEE: Local maxima-minima problems.

Second derivative test for local extremaEssential

f'(a)=0,\ f''(a)>0\Rightarrow f(a)\text{ local minimum};\qquad f'(a)=0,\ f''(a)<0\Rightarrow f(a)\text{ local maximum}

Variables: $a$ is a critical point.
Conditions: If $f''(a)=0$ , test is inconclusive.
Where used in JEE: Quick maxima-minima classification.

Necessary condition for interior extremumImportant

f\text{ differentiable and has local extremum at interior point }a\Rightarrow f'(a)=0

Conditions: Applies to interior points where differentiability holds.
Where used in JEE: Theory questions and locating candidates for extrema.

Absolute maxima and minima on closed intervalImportant

\text{On }[a,b],\ \text{absolute extrema occur among }a,b\text{ and critical points in }(a,b)

Conditions: Function should be continuous on $[a,b]$ .
Where used in JEE: Optimization on a finite interval.

Tangency slope and tangent-normal basicsSupplementary

m_{\text{tangent}}=\frac{dy}{dx},\qquad m_{\text{normal}}=-\frac{1}{dy/dx}

Variables: Slopes at a point on the curve.
Conditions: For normal slope, tangent slope should be nonzero and finite; vertical/horizontal cases handled separately.
Where used in JEE: Application of derivative to tangents, normals, monotonic behavior near points.

Frequently asked questions

What are the important Limit, Continuity and Differentiability formulas for JEE?

This page lists 68+ JEE-relevant Limit, Continuity and Differentiability formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Limit, Continuity and Differentiability formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Limit, Continuity and Differentiability, covering Real-valued functions, algebra of functions, Polynomial, rational, trigonometric, logarithmic and exponential functions, Inverse functions, Graphs of simple functions, and more.

How should I revise the Limit, Continuity and Differentiability formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Limit, Continuity and Differentiability MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Limit, Continuity and Differentiability?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.