Math · Calculus

Continuity and Differentiability formulas for JEE

Every Continuity and Differentiability formula you need for JEE, grouped by concept.

FormulasRevision notes
30 formulas6 concepts
01Continuity and Differentiability02Chain Rule03Derivatives of Special Functions04Parametric Differentiation05Second Order Derivatives06Mean Value Theorems
01

Continuity and Differentiability

3 formulas

Continuity at a Point

lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)

Definition of continuity of a function f at a point c.

applies whenFunction f must be defined at x = c.
continuitylimits

First Principle of Derivative

f′(c)=lim⁡h→0f(c+h)−f(c)hf'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}

Definition of derivative using limits.

applies whenLimit must exist.
derivativefirst-principlelimits

L'Hôpital's Rule

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Evaluating indeterminate limits.

applies whenLimit must yield 0/0 or \infty/\infty.
limitslhopitaljee-advanced
02

Chain Rule

8 formulas

Chain Rule

dfdx=dvdt⋅dtdx\frac{df}{dx} = \frac{dv}{dt} \cdot \frac{dt}{dx}

Derivative of a composite function.

applies whenBoth dt/dx and dv/dt must exist.
differentiationchain-rulecomposite

Leibniz Rule (Differentiation under Integral)

ddx∫a(x)b(x)f(t)dt=f(b(x))b′(x)−f(a(x))a′(x)\frac{d}{dx} \int_{a(x)}^{b(x)} f(t) dt = f(b(x))b'(x) - f(a(x))a'(x)

Differentiating an integral with variable limits.

applies whenIntegrand is continuous, bounds are differentiable.
differentiationintegraljee-advanced

Power Rule

ddxxn=nxn−1\frac{d}{dx} x^n = n x^{n-1}

Derivative of a polynomial term.

applies whenn is a real number.
differentiationpower-rule

Product Rule (Leibnitz)

(uv)′=u′v+uv′(uv)' = u'v + uv'

Derivative of the product of two functions.

applies whenu and v must be differentiable.
differentiationproduct-rule

Quotient Rule

(uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Derivative of the quotient of two functions.

applies whenv≠0v \neq 0
differentiationquotient-rule

Derivative of Cosine

ddx(cos⁡x)=−sin⁡x\frac{d}{dx}(\cos x) = -\sin x

Derivative of the cosine function.

applies whenx is in radians.
derivativestrigonometriccosine

Derivative of Sine

ddx(sin⁡x)=cos⁡x\frac{d}{dx}(\sin x) = \cos x

Derivative of the sine function.

applies whenx is in radians.
derivativestrigonometricsine

Derivative of Tangent

ddx(tan⁡x)=sec⁡2x\frac{d}{dx}(\tan x) = \sec^2 x

Derivative of the tangent function.

applies whenx is in radians, x≠(2n+1)π2x \neq \frac{(2n+1)\pi}{2}.
derivativestrigonometrictangent
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03

Derivatives of Special Functions

14 formulas

Derivative of sin^-1(2x/(1+x^2))

ddxsin⁡−1(2x1+x2)=21+x2\frac{d}{dx} \sin^{-1}\left(\frac{2x}{1+x^2}\right) = \frac{2}{1+x^2}

Derivative simplification using inverse trig substitution.

applies when∣x∣<1|x| < 1 for exact principal value identity
differentiationinverse-trigsubstitution

Derivative of tan^-1(sin x / (1+cos x))

ddxtan⁡−1(sin⁡x1+cos⁡x)=12\frac{d}{dx} \tan^{-1}\left(\frac{\sin x}{1+\cos x}\right) = \frac{1}{2}

Derivative simplification via half-angle formulas.

applies whencos⁡x≠−1\cos x \neq -1
differentiationinverse-trigsubstitution

Derivative of Inverse Function

g′(x)=1f′(g(x))g'(x) = \frac{1}{f'(g(x))}

Finding derivative of g(x) = f^-1(x).

applies whenf′(g(x))≠0f'(g(x)) \neq 0
differentiationinverse-functionsjee-advanced

Logarithm Base Change

log⁡ap=log⁡bplog⁡ba\log_a p = \frac{\log_b p}{\log_b a}

Formula to change the base of a logarithm.

applies whena,b>0a, b > 0, a,b≠1a, b \neq 1, p>0p > 0
logarithmidentities

Logarithmic Differentiation

dydx=[u(x)]v(x)[v(x)u′(x)u(x)+v′(x)log⁡u(x)]\frac{dy}{dx} = [u(x)]^{v(x)} \left[ v(x) \frac{u'(x)}{u(x)} + v'(x) \log u(x) \right]

Formula for differentiating variable base to variable power.

applies whenu(x)>0u(x) > 0
differentiationlogarithmic-differentiation

Logarithm Power Rule

log⁡b(pn)=nlog⁡bp\log_b (p^n) = n \log_b p

Logarithm of a variable raised to a power.

applies whenp>0p > 0
logarithmidentities

Logarithm Product Rule

log⁡b(pq)=log⁡bp+log⁡bq\log_b (pq) = \log_b p + \log_b q

Logarithm of a product.

applies whenp,q>0p, q > 0
logarithmidentities

Logarithm Quotient Rule

log⁡b(pq)=log⁡bp−log⁡bq\log_b \left(\frac{p}{q}\right) = \log_b p - \log_b q

Logarithm of a quotient.

applies whenp,q>0p, q > 0
logarithmidentities

Derivative of General Exponential

ddxax=axlog⁡a\frac{d}{dx} a^x = a^x \log a

Derivative of a^x.

applies whena>0a > 0
differentiationexponential

Derivative of Exponential

ddxex=ex\frac{d}{dx} e^x = e^x

Derivative of natural exponential function.

applies whenx∈Rx \in \mathbb{R}
differentiationexponential

Derivative of Inverse Cosine

ddxcos⁡−1x=−11−x2\frac{d}{dx} \cos^{-1} x = \frac{-1}{\sqrt{1-x^2}}

Standard derivative of arccos(x).

applies whenx∈(−1,1)x \in (-1, 1)
differentiationinverse-trig

Derivative of Inverse Sine

ddxsin⁡−1x=11−x2\frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1-x^2}}

Standard derivative of arcsin(x).

applies whenx∈(−1,1)x \in (-1, 1)
differentiationinverse-trig

Derivative of Inverse Tangent

ddxtan⁡−1x=11+x2\frac{d}{dx} \tan^{-1} x = \frac{1}{1+x^2}

Standard derivative of arctan(x).

applies whenx∈Rx \in \mathbb{R}
differentiationinverse-trig

Derivative of Natural Logarithm

ddxlog⁡x=1x\frac{d}{dx} \log x = \frac{1}{x}

Derivative of natural logarithm base e.

applies whenx>0x > 0
differentiationlogarithm
04

Parametric Differentiation

1 formula

Parametric Differentiation

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

First derivative of parametric equations.

applies whendx/dt≠0dx/dt \neq 0
differentiationparametric
05

Second Order Derivatives

2 formulas

Second Order Parametric Derivative

d2ydx2=ddt(dy/dtdx/dt)⋅dtdx\frac{d^2y}{dx^2} = \frac{d}{dt}\left(\frac{dy/dt}{dx/dt}\right) \cdot \frac{dt}{dx}

Calculating second derivative for parametric equations.

applies whendx/dt≠0dx/dt \neq 0
differentiationparametrichigher-orderjee-advanced

Second Order Derivative

d2ydx2=ddx(dydx)\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)

Basic definition of second order derivative.

applies whenFirst derivative must be differentiable.
differentiationhigher-order
06

Mean Value Theorems

2 formulas

Lagrange's Mean Value Theorem (LMVT)

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}

Formula for LMVT.

applies whenf is continuous on [a,b] and differentiable on (a,b).
theoremsmvtjee-advanced

Rolle's Theorem

f′(c)=0f'(c) = 0

Condition for existence of horizontal tangent.

applies whenf is continuous on [a,b], differentiable on (a,b), and f(a)=f(b).
theoremsmvtjee-advanced
Other chapters
MathSets34 formulasMathRelations and Functions18 formulasMathTrigonometric Functions56 formulasMathComplex Numbers and Quadratic Equations36 formulasMathPermutations and Combinations21 formulasMathBinomial Theorem24 formulasMathSequence and Series25 formulasMathStraight Lines27 formulasMathConic Sections35 formulasMathIntroduction to Three-dimensional Geometry9 formulasMathLimits and Derivatives32 formulasMathStatistics25 formulasMathProbability17 formulasMathRelations and Functions14 formulasMathInverse Trigonometric Functions33 formulasMathMatrices44 formulasMathDeterminants23 formulasMathApplications of Derivatives26 formulasMathIntegrals47 formulasMathApplications of the Integrals14 formulasMathDifferential Equations23 formulasMathVectors29 formulasMathThree-dimensional Geometry32 formulasMathProbability15 formulas

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