Math · Calculus

Limits and Derivatives formulas for JEE

Every Limits and Derivatives formula you need for JEE, grouped by concept.

32 formulas4 concepts

01Limits 02Derivatives 03Rules of Differentiation 04Derivatives of Standard Functions

Limits

17 formulas

Limit of (1-cos(x))/x

\lim_{x \to 0} \frac{1 - \cos x}{x} = 0

Standard limit of cosine expression as angle approaches zero.

applies when

x

must be measured in radians.

limitstrigonometriccosine

Limit Definition Notation

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

The expected value of f(x) as x approaches a is l.

applies whenLeft hand limit equals right hand limit.

limitsnotation

Limit of Exponential Function

\lim_{x \to 0} \frac{e^x - 1}{x} = 1

Standard JEE-Advanced limit involving the exponential function.

limitsexponentialjee-advanced

L'Hôpital's Rule

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Method for evaluating indeterminate limit forms by taking derivatives.

applies whenLimit evaluates to

0/0

\infty/\infty

limitsindeterminate_formsl_hopitaljee-advanced

Limit of Logarithmic Function

\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1

Standard JEE-Advanced limit involving the natural logarithm.

limitslogarithmicjee-advanced

1^Infinity Indeterminate Form Limit

\lim_{x \to a} f(x)^{g(x)} = e^{\lim_{x \to a} g(x)[f(x) - 1]}

Formula for solving limits of the $1^{\infty}$ indeterminate form.

applies when

\lim f(x) = 1

and

\lim g(x) = \infty

limitsindeterminate_formsjee-advanced

Standard Limit for Powers

\lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}

Standard formula for evaluating limits of power functions.

applies whenn is a rational number, a is positive.

limitsalgebrapower

Limit Difference Rule

\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)

The limit of a difference is the difference of the limits.

applies whenBoth individual limits must exist.

limitsalgebradifference

Limit Product Rule

\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

The limit of a product is the product of the limits.

applies whenBoth individual limits must exist.

limitsalgebraproduct

Limit Quotient Rule

\lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}

The limit of a quotient is the quotient of the limits.

applies whenBoth limits exist and

\lim_{x \to a} g(x) \neq 0

limitsalgebraquotient

Limit Scalar Multiple Rule

\lim_{x \to a} [\lambda \cdot f(x)] = \lambda \cdot \lim_{x \to a} f(x)

Constants can be factored out of limit evaluations.

applies whenLimit of f(x) exists,

\lambda

is a real number.

limitsalgebrascalar

Limit Sum Rule

\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

The limit of a sum is the sum of the limits.

applies whenBoth individual limits must exist.

limitsalgebrasum

Limit of sin(x)/x

\lim_{x \to 0} \frac{\sin x}{x} = 1

Standard limit of the sine function as angle approaches zero.

applies when

x

must be measured in radians.

limitstrigonometricsine

Fundamental Trigonometric Inequality

\sin x < x < \tan x

Geometric inequality used to prove trigonometric limits.

applies when

x

is in radians and

0 < x < \frac{\pi}{2}

limitstrigonometricinequalitysandwich

Limit of tan(x)/x

\lim_{x \to 0} \frac{\tan x}{x} = 1

Derived limit for the tangent function.

applies when

x

must be measured in radians.

limitstrigonometrictangent

Half-Angle Sine Identity

1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right)

Trigonometric identity used to simplify 1-cos(x) in limits.

trigonometryidentityhalf_angle

Sum-to-Product Sine Identity

\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)

Identity used to evaluate the derivative of sine by first principles.

trigonometryidentitysum_to_product

Derivatives

2 formulas

Average Velocity

v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}

Average rate of change of distance with respect to time.

applies when

t_2 \neq t_1

velocityaveragerate_of_change

Derivative from First Principle

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

Definition of the derivative of a function at a point x.

applies whenLimit must exist.

derivativesdefinitionfirst_principle

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Rules of Differentiation

5 formulas

Product Rule (Leibnitz Rule)

(uv)' = u'v + uv'

Rule for differentiating the product of two functions.

applies whenu and v must be differentiable functions.

derivativesrulesalgebraproduct

Quotient Rule

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

Rule for differentiating the quotient of two functions.

applies whenu and v are differentiable functions and

v \neq 0

derivativesrulesalgebraquotient

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Rule for differentiating composite functions.

applies whenf and g must be differentiable functions.

derivativesrulescompositejee-advanced

Derivative Difference Rule

(u - v)' = u' - v'

Derivative of a difference is the difference of derivatives.

applies whenu and v must be differentiable functions.

derivativesrulesalgebradifference

Derivative Sum Rule

(u + v)' = u' + v'

Derivative of a sum is the sum of derivatives.

applies whenu and v must be differentiable functions.

derivativesrulesalgebrasum

Derivatives of Standard Functions

8 formulas

Derivative of Cosine

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

Derivative of the cosine function.

applies whenx is in radians.

derivativestrigonometriccosine

Derivative of Cosecant

\frac{d}{dx}(\text{cosec } x) = -\text{cosec } x \cot x

Derivative of the cosecant function.

applies whenx is in radians,

x \neq n\pi

derivativestrigonometriccosecant

Derivative of Cotangent

\frac{d}{dx}(\cot x) = -\text{cosec}^2 x

Derivative of the cotangent function.

applies whenx is in radians,

x \neq n\pi

derivativestrigonometriccotangent

Derivative of a Polynomial

\frac{d}{dx}(a_n x^n + \dots + a_1 x + a_0) = n a_n x^{n-1} + \dots + a_1

Direct application of power, sum, and scalar rules on polynomials.

derivativesalgebrapolynomial

Power Rule

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

Standard derivative of a polynomial or power function.

applies whenn is any real number.

derivativesalgebrapower

Derivative of Secant

\frac{d}{dx}(\sec x) = \sec x \tan x

Derivative of the secant function.

applies whenx is in radians,

x \neq \frac{(2n+1)\pi}{2}

derivativestrigonometricsecant

Derivative of Sine

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

Derivative of the sine function.

applies whenx is in radians.

derivativestrigonometricsine

Derivative of Tangent

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

Derivative of the tangent function.

applies whenx is in radians,

x \neq \frac{(2n+1)\pi}{2}

derivativestrigonometrictangent

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