Maths · Class 12 · Limit, Continuity and Differentiability

Application of Derivatives for JEE

Q: Is Application of Derivatives important for JEE Main 2027?

Appears frequently. It is part of the official 'Limit, Continuity and Differentiability' unit in the JEE Main 2027 syllabus.

Q: What are the most important formulas in Application of Derivatives?

Tangent: y − y₁ = m(x − x₁), m = f′(x₁); Normal slope = −1/m; LMVT: f′(c) = (f(b) − f(a))/(b − a).

Q: How should I study Application of Derivatives for JEE?

Understand the key ideas from NCERT or one full lecture, learn the formula sheet, then solve 25–30 past JEE questions on the topic. Maxima–minima word problems need a clean function first. Set it up carefully.

Everything you need for this topic on one page: the ideas, the formulas, the traps, and free lectures that explain it well.

Last updated: 26 October 2026

Direct answer

Rate of change, increasing and decreasing functions, tangents and normals, maxima and minima, and the mean value theorems.

Key ideas

Application of Derivatives: what you must understand

f′(x) > 0 means increasing
At a local max f′ = 0 and f″ < 0
Check endpoints for absolute max/min on a closed interval
Rolle's and Lagrange's mean value theorems

Formula sheet

Must-know formulas for Application of Derivatives

Tangent: y − y₁ = m(x − x₁), m = f′(x₁)
Normal slope = −1/m
LMVT: f′(c) = (f(b) − f(a))/(b − a)
Second derivative test: f″(c) < 0 → max

One-page formula sheet

Printable Application of Derivatives formula sheet: formulas, key ideas and traps on one page. Save it as a PDF for offline revision.

Avoid

Common mistakes students make

Forgetting endpoints
Using f′ = 0 points without checking sign change
Sign errors in normal slope

For the exam

How this topic is asked in JEE

Appears frequently. Maxima–minima word problems need a clean function first. Set it up carefully.

Prerequisites: Continuity and Differentiability.

NCERT map

Where Application of Derivatives is in NCERT

Chapter names follow the current rationalised NCERT textbooks. Read these sections before solving PYQs.

Class	Book	Chapter	Sections JEE asks from
Class 12	Mathematics Part I	Application of Derivatives	Rate of Change of Quantities; Increasing and Decreasing Functions; Tangents and Normals; Maxima and Minima

JEE Advanced level

A JEE Advanced-level example on Application of Derivatives

Question. A curve is defined by y = x³ - 3x² + 2x. A tangent is drawn to this curve at a point P where the tangent makes an angle of 45° with the positive x-axis. If P lies in the first quadrant, find the equation of the normal at P.

Show solution

Find dy/dx = 3x² - 6x + 2. Since tangent makes 45° with x-axis, slope = tan(45°) = 1.
Solve 3x² - 6x + 2 = 1, giving 3x² - 6x + 1 = 0. Using quadratic formula: x = (3 ± √6)/3.
Check which x gives a point in first quadrant. Write y = x(x-1)(x-2). For x = (3-√6)/3 ≈ 0.184, we get y > 0. For x = (3+√6)/3 ≈ 1.816, we get y < 0. So x = (3-√6)/3.
Calculate y-coordinate: y = [(3-√6)/3] × [(-√6)/3] × [(-3-√6)/3] = √6/9. Point P = ((3-√6)/3, √6/9).
Slope of normal = -1 (negative reciprocal of tangent slope). Equation: y - √6/9 = -1(x - (3-√6)/3). Simplify to get 9x + 9y = 9 - 2√6.

Answer: 9x + 9y = 9 - 2√6 (or equivalently x + y = 1 - (2√6)/9)

This example was solved twice independently and both solutions agree.

Free lectures

Best free lectures for Application of Derivatives

Full-chapter free lectures picked from YouTube search results by views, from established JEE channels (mostly in Hinglish). Use the subtopic list to jump straight to the part you need (timestamps come from each video's own chapter list). Watch one, then solve PYQs the same day.

APPLICATION OF DERIVATIVES in One Shot: All Concepts & PYQs Covered | JEE Main & Advanced — JEE Wallah, 5:33:55
Jump to a subtopic (16)
- 00:00 Introduction
- 02:25 Derivative as a Rate of Change (Questions)
- 31:40 Monotonicity at a Point & Test for Monotonicity
- 41:05 Questions
- 01:30:10 Maxima & Minima
- 01:52:52 First Derivative Test (Questions)
- 02:26:28 Second Derivative Test (Questions)
- 03:17:42 Concavity & Point of Inflection
- 03:28:52 Questions
- 03:40:03 Graph for Cubic Expression
- 04:23:38 Tangent & Normal at a Point x=a
- 04:38:46 Tangent & Normal from an Exterior Point
- 04:42:44 Angle Between 2 Curves
- 04:59:00 Min Distance between 2 Curves
- 05:23:25 Lagrange's Mean Value Theorem (Questions)
- 05:32:42 Thank you
Application of Derivatives Class 12 | Tangents & Normals, Mean Value Theorems | JEE Main & Advanced — JEE Nexus by Unacademy, 4:01:13
Jump to a subtopic (11)
- 0:00 Introduction and Nature of Chapter
- 2:29 Index
- 5:52 Brief Introduction of tangents and Normal
- 7:53 Tangent/Normal at a given point on the curve
- 52:29 Tangent/ Normal having given slope
- 1:41:43 Tangent/Normal from an external Point
- 2:03:09 Proving some general statements
- 2:29:29 Angle of Intersection between the curves
- 2:50:31 Length of tangent, normal, subtangent and subnormal
- 3:03:19 Mean Value theorem (Rolle's Theorem)
- 3:35:59 Lagrange's Mean value theorem

Practise

Where to practise for free

3 JEE Main 2026 PYQs on Application of Derivatives, with solutions.
Read the chapter in NCERT textbooks (official, free), then solve its NCERT Exemplar (official, free) questions.
Chapter-wise past questions: JEE Main and Advanced PYQs and question bank.
Timed practice tests: NTA Abhyas practice tests (official, free) and our free mock tests.
See it move: interactive simulation.

FAQs

Frequently asked questions

Is Application of Derivatives important for JEE Main 2027?