JEE · Mathematics

Differentiation

Understand the derivative as a limit-defined local rate, differentiate standard and composite functions with correct domains, and distinguish differentiability from continuity.

Subject: Mathematics
Syllabus unit: Differentiation

Main 2026 and Advanced 2026 both include standard derivative rules
Every standard derivative carries its domain condition
No invented weightage, question counts or trend percentages

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In short

The derivative measures local change and is defined by a limit of difference quotients when that limit exists.

Rules such as product, quotient and chain rule are valid only where the component functions and required denominators are defined and differentiable. Differentiability at a point implies continuity there, but continuity does not by itself imply differentiability.

Syllabus mapping

Syllabus mapping
Unit	Topics
Differentiation	Derivative definition as a limit, Sum, difference, product and quotient rules, Chain rule, Standard trigonometric, inverse-trigonometric, exponential and logarithmic derivatives, Implicit differentiation up to order two, Differentiability versus continuity

Unit
Differentiation
Topics
Derivative definition as a limit, Sum, difference, product and quotient rules, Chain rule, Standard trigonometric, inverse-trigonometric, exponential and logarithmic derivatives, Implicit differentiation up to order two, Differentiability versus continuity

What this chapter contains and why it matters

What this chapter contains and why it matters
Question	Direct answer
What is the chapter about?	The derivative as a limit-defined local rate, standard and composite differentiation rules, their domains, and the differentiability-continuity relationship.
What is the central method choice?	Identify the function's domain and structure, apply the smallest valid rule set, and check denominator and domain conditions before executing.
Where do most mistakes begin?	Treating continuity and differentiability as equivalent, omitting the chain-rule inner derivative, and forgetting inverse-trig derivative domains.
What should come before this chapter?	Limits and Continuity, and Inverse Trigonometry for branch and domain rules.
What comes after it?	Application of Derivatives uses differentiation results for monotonicity, extrema and rates; Integration reverses the differentiation process.

Question
What is the chapter about?
Direct answer
The derivative as a limit-defined local rate, standard and composite differentiation rules, their domains, and the differentiability-continuity relationship.
Question
What is the central method choice?
Direct answer
Identify the function's domain and structure, apply the smallest valid rule set, and check denominator and domain conditions before executing.
Question
Where do most mistakes begin?
Direct answer
Treating continuity and differentiability as equivalent, omitting the chain-rule inner derivative, and forgetting inverse-trig derivative domains.
Question
What should come before this chapter?
Direct answer
Limits and Continuity, and Inverse Trigonometry for branch and domain rules.
Question
What comes after it?
Direct answer
Application of Derivatives uses differentiation results for monotonicity, extrema and rates; Integration reverses the differentiation process.

The official JEE documents define content scope. They do not publish chapter weightage, so none is asserted here.

Official JEE syllabus mapping for Differentiation

Official JEE syllabus mapping for Differentiation
Concept group	JEE Main 2026	JEE Advanced 2026	Preparation note
Differentiability and standard rules	Explicitly includes differentiability; sum, difference, product and quotient rules.	Explicitly includes derivatives of sums, differences, products and quotients.	Both authorities require the same structural rules with their conditions.
Standard function derivatives	Explicitly includes trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit differentiation up to order two.	Explicitly includes polynomial, rational, trigonometric, inverse-trigonometric, exponential and logarithmic derivatives, and implicit derivatives up to order two.	Learn each standard derivative with its exact domain condition.
Logarithmic differentiation	Not separately named as a syllabus bullet.	Not separately named as a syllabus bullet.	Treat as supporting mathematics, not an explicit official inclusion.

Concept group
Differentiability and standard rules
JEE Main 2026
Explicitly includes differentiability; sum, difference, product and quotient rules.
JEE Advanced 2026
Explicitly includes derivatives of sums, differences, products and quotients.
Preparation note
Both authorities require the same structural rules with their conditions.
Concept group
Standard function derivatives
JEE Main 2026
Explicitly includes trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit differentiation up to order two.
JEE Advanced 2026
Explicitly includes polynomial, rational, trigonometric, inverse-trigonometric, exponential and logarithmic derivatives, and implicit derivatives up to order two.
Preparation note
Learn each standard derivative with its exact domain condition.
Concept group
Logarithmic differentiation
JEE Main 2026
Not separately named as a syllabus bullet.
JEE Advanced 2026
Not separately named as a syllabus bullet.
Preparation note
Treat as supporting mathematics, not an explicit official inclusion.

JEE Main 2026 and JEE Advanced 2026 are independent authorities.

Before this chapter

Concepts in this chapter

1. Start from the limit definition

f'(a) is the limit of a difference quotient when that limit exists.

The derivative at a is f'(a)=lim[h->0](f(a+h)-f(a))/h if the finite real limit exists in the standard interior-point setting.

2. Detect the function's structure before choosing a rule

Identify sum, product, quotient or composition structure, then apply the smallest valid rule set.

Identify the function's real domain first, then detect sum, product, quotient or composition structure, and apply the smallest valid rule set.

3. Require the denominator to be nonzero

The quotient rule requires the denominator function to be nonzero at the differentiation point.

For a quotient, the denominator must be nonzero at the point of differentiation. Applying the quotient rule where the denominator is zero produces an invalid result.

4. Enforce inverse-trig and log domains before differentiating

Inverse-trig and log functions each carry their own domain, which must hold before their derivative is used.

For inverse-trig and log functions, enforce their domains before differentiating, since the resulting derivative expression is valid only within that domain.

5. Treat y as a function of x for implicit relations

In an implicit relation F(x,y)=0, differentiate both sides and attach dy/dx to every y-derivative.

For implicit differentiation, treat y as a function of x and attach dy/dx when differentiating any term containing y.

6. Keep differentiability and continuity as separate properties

Differentiability implies continuity, but continuity does not imply differentiability.

If f is differentiable at a, f is continuous at a in the standard real-variable setting. The converse is false; |x| at 0 is a standard counterexample.

Mathematical object definitions

Mathematical object definitions
Object	Meaning
Derivative at a	f'(a)=lim[h->0](f(a+h)-f(a))/h if the finite real limit exists in the standard interior-point setting.
Differentiable	Derivative exists at the point or throughout the stated interval.
First derivative	Local rate/slope function f'.
Second derivative	Derivative of f' where it exists.
Composite function	f(g(x)); differentiated by chain rule.
Implicit relation	Relation F(x,y)=0 where y is not necessarily isolated before differentiation.

Object
Derivative at a
Meaning
f'(a)=lim[h->0](f(a+h)-f(a))/h if the finite real limit exists in the standard interior-point setting.
Object
Differentiable
Meaning
Derivative exists at the point or throughout the stated interval.
Object
First derivative
Meaning
Local rate/slope function f'.
Object
Second derivative
Meaning
Derivative of f' where it exists.
Object
Composite function
Meaning
f(g(x)); differentiated by chain rule.
Object
Implicit relation
Meaning
Relation F(x,y)=0 where y is not necessarily isolated before differentiation.

Method selector

Match the function structure to the correct differentiation rule.

Method selector
Signal	First method	Required check
Sum/product/quotient	Structural rule	Component differentiability and nonzero denominator
Nested function	Chain rule, outside first then inside	Every layer differentiated
Implicit relation	Differentiate both sides with y=y(x)	dy/dx attached to every y-term
Inverse trig	Branch/domain check then standard derivative	Domain of the inverse-trig function
Cusp/absolute value/piecewise	Left and right derivatives or definition	One-sided derivatives compared
Second derivative	Differentiate first derivative only where it remains differentiable	First derivative differentiability

Signal
Sum/product/quotient
First method
Structural rule
Required check
Component differentiability and nonzero denominator
Signal
Nested function
First method
Chain rule, outside first then inside
Required check
Every layer differentiated
Signal
Implicit relation
First method
Differentiate both sides with y=y(x)
Required check
dy/dx attached to every y-term
Signal
Inverse trig
First method
Branch/domain check then standard derivative
Required check
Domain of the inverse-trig function
Signal
Cusp/absolute value/piecewise
First method
Left and right derivatives or definition
Required check
One-sided derivatives compared
Signal
Second derivative
First method
Differentiate first derivative only where it remains differentiable
Required check
First derivative differentiability

Worked reasoning: continuity without differentiability

One-sided difference quotients disagree at a cusp

Consider f(x)=|x| at x=0. f(0)=0, and |x| is continuous at 0. Is it differentiable at 0?

For h>0: (|h|-0)/h=1. For h<0: (|h|-0)/h=-1. Left and right derivative limits are different, so f'(0) does not exist.

This is a direct counterexample to the false statement that continuity implies differentiability.

Formula and theorem records with conditions

Formula and theorem records with conditions
Record	Expression	Conditions
Definition	f'(x)=lim[h->0](f(x+h)-f(x))/h	Limit exists; x is a valid local domain point
Sum/difference	(f±g)'=f'±g'	f,g differentiable
Product	(fg)'=f'g+fg'	f,g differentiable
Quotient	(f/g)'=(f'g-fg')/g²	f,g differentiable and g≠0
Chain rule	(f∘g)'(x)=f'(g(x))g'(x)	g differentiable at x; f differentiable at g(x)
Polynomial power	d(x^n)/dx=nx^(n-1)	n nonnegative integer for all real x
Sine	(sinx)'=cosx	x in radians
Cosine	(cosx)'=-sinx	radians
Tangent	(tanx)'=sec²x	cosx≠0
Cotangent	(cotx)'=-csc²x	sinx≠0
Exponential	(e^x)'=e^x	all real x
General positive-base exponential	(a^x)'=a^x ln a	a>0
Natural log	(lnx)'=1/x	x>0
Base-a logarithm	(log_a x)'=1/(x ln a)	x>0, a>0, a≠1
Inverse sine	(sin⁻¹x)'=1/√(1-x²)	\|x\|<1 for finite real derivative
Inverse cosine	(cos⁻¹x)'=-1/√(1-x²)	\|x\|<1
Inverse tangent	(tan⁻¹x)'=1/(1+x²)	all real x
Inverse cotangent	(cot⁻¹x)'=-1/(1+x²)	all real x under the stated NCERT principal branch
Inverse secant	(sec⁻¹x)'=1/(\|x\|√(x²-1))	\|x\|>1
Inverse cosecant	(csc⁻¹x)'=-1/(\|x\|√(x²-1))	\|x\|>1
Differentiability implies continuity	If f is differentiable at a, f is continuous at a	Standard real-variable setting
Converse warning	Continuity does not imply differentiability	Counterexample \|x\| at 0

Record
Definition
Expression
f'(x)=lim[h->0](f(x+h)-f(x))/h
Conditions
Limit exists; x is a valid local domain point
Record
Sum/difference
Expression
(f±g)'=f'±g'
Conditions
f,g differentiable
Record
Product
Expression
(fg)'=f'g+fg'
Conditions
f,g differentiable
Record
Quotient
Expression
(f/g)'=(f'g-fg')/g²
Conditions
f,g differentiable and g≠0
Record
Chain rule
Expression
(f∘g)'(x)=f'(g(x))g'(x)
Conditions
g differentiable at x; f differentiable at g(x)
Record
Polynomial power
Expression
d(x^n)/dx=nx^(n-1)
Conditions
n nonnegative integer for all real x
Record
Sine
Expression
(sinx)'=cosx
Conditions
x in radians
Record
Cosine
Expression
(cosx)'=-sinx
Conditions
radians
Record
Tangent
Expression
(tanx)'=sec²x
Conditions
cosx≠0
Record
Cotangent
Expression
(cotx)'=-csc²x
Conditions
sinx≠0
Record
Exponential
Expression
(e^x)'=e^x
Conditions
all real x
Record
General positive-base exponential
Expression
(a^x)'=a^x ln a
Conditions
a>0
Record
Natural log
Expression
(lnx)'=1/x
Conditions
x>0
Record
Base-a logarithm
Expression
(log_a x)'=1/(x ln a)
Conditions
x>0, a>0, a≠1
Record
Inverse sine
Expression
(sin⁻¹x)'=1/√(1-x²)
Conditions
|x|<1 for finite real derivative
Record
Inverse cosine
Expression
(cos⁻¹x)'=-1/√(1-x²)
Conditions
|x|<1
Record
Inverse tangent
Expression
(tan⁻¹x)'=1/(1+x²)
Conditions
all real x
Record
Inverse cotangent
Expression
(cot⁻¹x)'=-1/(1+x²)
Conditions
all real x under the stated NCERT principal branch
Record
Inverse secant
Expression
(sec⁻¹x)'=1/(|x|√(x²-1))
Conditions
|x|>1
Record
Inverse cosecant
Expression
(csc⁻¹x)'=-1/(|x|√(x²-1))
Conditions
|x|>1
Record
Differentiability implies continuity
Expression
If f is differentiable at a, f is continuous at a
Conditions
Standard real-variable setting
Record
Converse warning
Expression
Continuity does not imply differentiability
Conditions
Counterexample |x| at 0

Logarithmic differentiation is a supporting technique, not a separately named 2026 Main or Advanced syllabus bullet; a valid ln|y| formulation for nonzero y preserves real logarithm conditions.

Common mistakes and what they actually indicate

Treating continuity and differentiability as equivalent.
Knowledge gap
Why it happens
Differentiability implies continuity, but continuity does not imply differentiability; |x| at 0 is a standard counterexample.
How it is corrected
Check the derivative definition directly at a cusp or piecewise-join point instead of assuming continuity is enough.
Omitting the chain-rule inner derivative.
Execution error
Why it happens
The chain rule requires multiplying by the derivative of the inner function, not just the outer derivative.
How it is corrected
Differentiate the outer function first, then multiply by the derivative of the inner function.
Using the quotient rule where the denominator is zero.
Execution error
Why it happens
The quotient rule requires g≠0 at the point of differentiation.
How it is corrected
Check the denominator function's value before applying the quotient rule.
Forgetting inverse-trig derivative domains.
Recall gap
Why it happens
Each inverse-trig derivative expression, such as 1/√(1-x²), is valid only within its stated domain.
How it is corrected
Check the domain condition for the specific inverse-trig derivative before using the formula.
Forgetting radians in trig derivative derivations or formulas.
Execution error
Why it happens
The standard trig derivatives such as (sinx)'=cosx assume x is in radians.
How it is corrected
Confirm the angle variable is in radians before applying a standard trig derivative.
Treating an implicit y term as constant.
Decision / selection error
Why it happens
In an implicit relation, y is a function of x, so any term containing y needs dy/dx when differentiated.
How it is corrected
Attach dy/dx to every derivative of a y-term when differentiating implicitly.
Differentiating through a cusp without checking one-sided derivatives.
Decision / selection error
Why it happens
A symbolic rule can hide a point where the left and right derivatives disagree.
How it is corrected
Use the derivative definition and compare one-sided derivatives at a suspected cusp or piecewise join.
Using logarithmic differentiation without a valid real log interval.
Execution error
Why it happens
Logarithmic differentiation requires the logged expression to be positive, or a valid ln|y| formulation for nonzero y.
How it is corrected
Confirm the real logarithm condition holds on the working interval before using logarithmic differentiation.

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