If you are working out how to study Application of Derivatives for JEE, start with two facts. JEE Main asked 51 questions from this chapter across 2024, 2025 and 2026, and JEE Advanced asked 15 questions from it between 2015 and 2026. This guide breaks every one of those questions down by sub-topic, based on the papers conducted by the National Testing Agency (jeemain.nta.ac.in) for JEE Main and the JEE Advanced papers from 2015 to 2026.
We sorted each question into one of five groups: maxima and minima of a given function, monotonicity, critical points and counting (extrema or roots), geometric optimisation, and rate of change. The tables below show exactly where the marks have come from. The rest of the guide shows how to win them, one sub-topic at a time.
There is also a syllabus change that most students have not caught up with. From 2024, the JEE Main syllabus no longer lists tangents and normals, or Rolle's and Lagrange's mean value theorems. Many students still spend weeks on tangent and normal problems from older books. For JEE Main, that time is better spent on maxima, minima and monotonicity, which is where the questions now are.
It is a steady source of marks in every JEE Main session. JEE Main had 26 questions from this chapter in 2024, 14 in 2025 and 11 in 2026. That is 4.33%, 2.95% and 2.32% of all Maths questions in those years. The share has fallen since 2024, but the chapter still appeared in both sessions of every year, and its questions are among the most predictable in Calculus.
The syllabus is now narrow, so the questions repeat. With tangents, normals, Rolle's theorem and the mean value theorem removed from JEE Main, almost every question from 2024 to 2026 asks one of three things: find the maximum or minimum value, find where a function increases or decreases, or count critical points, extrema or roots. A chapter with three question types is a chapter you can master.
It feeds other chapters. Counting the roots of an equation, finding the range of a function, proving an inequality and sketching a curve for area under curves all use the same derivative tests. Students who are comfortable with this chapter solve those questions faster in other chapters too.
JEE Advanced uses it for multi-correct questions. 8 of the 15 Advanced questions from 2015 to 2026 were "one or more correct", where each option is a separate statement about a function's extrema or monotonicity. These reward clean, careful reasoning more than heavy calculation.
"In this chapter, the derivative is only the first line of the solution. The marks are in what comes after: the sign chart, the end points, the points where the derivative does not exist. Students who draw the sign chart every time stop losing marks here." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
Real result · JEE 2026
Maxima, minima and monotonicity are routine marks once someone has checked your sign charts a few times. Learn this chapter 1-on-1 with an IITian Maths HOD and stop losing marks to end points and missed critical points.
Application of Derivatives packs
✔ Free 30-min mentorship call before you pay · ✔ 100% refund guarantee after the first session · ✔ 7-day WhatsApp access to your faculty
Any plan for how to study Application of Derivatives for JEE should start from the papers. Here is how many questions JEE Main asked from this chapter in each of the last three years.
| Year | Questions | MCQ + Numerical | Share of Maths Qs |
|---|---|---|---|
| 2024 | 26 | 20 + 6 | 4.33% |
| 2025 | 14 | 12 + 2 | 2.95% |
| 2026 | 11 | 8 + 2 (+1)* | 2.32% |
| Total | 51 | 40 + 10 (+1)* | — |
*One 2026 question is counted in the year total but could not be located in the public archive, so its type is not known.
And here are the same 51 questions sorted by what they actually test.
| Sub-topic | 2024 | 2025 | 2026 | Total |
|---|---|---|---|---|
| Maxima and minima of a given function (local and absolute values, building a polynomial from its extrema) | 8 | 5 | 5 | 18 |
| Monotonicity (increasing and decreasing intervals, statements about abstract functions) | 8 | 2 | 4 | 14 |
| Critical points and counting (number of critical points, local maxima or minima, real roots) | 5 | 3 | 1 | 9 |
| Geometric optimisation (largest rectangle, parallelogram or triangle, shortest distance) | 2 | 3 | 0 | 5 |
| Rate of change | 0 | 1 | 0 | 1 |
| Other, not classified or not located* | 3 | 0 | 1 | 4 |
| Total | 26 | 14 | 11 | 51 |
*2024: one question on the derivative of an inverse function, one question whose archived text did not render fully, and one area-under-a-parabola question that the archive files under this chapter but belongs to Area under Curves. 2026: the one question we could not locate. These are counted in the yearly totals only. Questions that test two ideas are counted once, under the idea they mainly test.
Two sub-topics dominate. Maxima and minima (18) and monotonicity (14) together gave 32 of the 51 questions, close to two-thirds of the chapter. Add the counting questions and you have 41 of 51, about 80%. Geometric optimisation is small but steady, and rate of change appeared only once. Tangents and normals did not appear at all, which matches the syllabus change.
Pattern 1: "Find the maximum and minimum on an interval" with a twist. 2026 asked for the minimum of f(x) = x2025 − x2000 on [0, 1], where the large powers scare students but the method is the usual one: set f'(x) = 0, solve for x, then compare with the end points. 2025 asked for the sum of the absolute maximum and minimum of a function on an interval, and 2024 did the same for a cubic on [−4, 4]. End points decide many of these answers.
Pattern 2: Build the polynomial from its extrema. 2025 gave a degree-four polynomial with extreme values at x = 4 and x = 5 and asked for f(2). 2026 did the same with a degree-five polynomial. 2024 asked for the values of a parameter that put a local maximum at a given point. The method is to write f'(x) from the given roots, integrate, and use the remaining conditions to fix the constants.
Pattern 3: Counting is the favourite numerical. The number of critical points (2026), the number of local maxima of a trigonometric function on an interval (2024), m − n for local minima and maxima (2025) and the number of local extrema of a function defined by an integral (2025) all appeared as MCQs. In numerical form, 2024 asked for the set of p for which f(x) = (p2 − 6p + 8)(sin22x − cos22x) + 2(2 − p)x + 7 has no critical point. Counting questions are fast marks once you can draw a sign chart.
Pattern 4: Monotonicity of functions you never see. Several questions give a function only through conditions: a twice-differentiable f with a bound on f'' (2026), g(x) = 3f(x/3) + f(3 − x) with f'' > 0 (2024), or 5f(x) + 4f(1/x) = x2 − 2 (2024). You must decide where g increases or decreases without a formula. The tool is always the same: differentiate g, then use the given sign information about f' or f''.
Pattern 5: Geometry optimisation came back in 2025. 2025 asked for the largest axis-parallel rectangle inside a region, the shortest distance between a parabola and a circle, and, as a numerical, the maximum area of a parallelogram with its vertices on the sides of a triangle. 2024 asked for the maximum area of a triangle and of a rectangle placed inside another rectangle. These questions are about setting up one variable correctly. The calculus is the easy part.
This is the 12-year view that most guides skip. JEE Advanced asked this chapter in 8 of the 12 years from 2015 to 2026.
| Year | Qs | What was asked |
|---|---|---|
| 2026 | 1 | f(x) = √x ln x − x + 1 on (0, ∞): decide whether f has a local maximum, a local minimum or neither, and whether f' is decreasing on (0, 1) (single correct) |
| 2025 | 1 | Piecewise-defined f on ℝ: statements about its local extrema, including at the joining point (multi-correct) |
| 2024 | 0 | — |
| 2023 | 0 | — (one 3D-geometry question filed under this chapter in the archive is not counted) |
| 2022 | 1 | g(x) = 2αx + 2α(1−x) with α defined by an infinite sine series: maximum and minimum of g and where they occur (multi-correct) |
| 2021 | 0 | — |
| 2020 | 1 | Rectangles under a curve with one side on the x-axis: area of the rectangle with the maximum perimeter (single correct) |
| 2019 | 2 | F(x) = ∫0x (t − 1)(t − 2)(t − 5) dt: local maxima and minima of F (multi-correct) · f(x) = sin(πx)/x2: comparing its successive local maxima and minima (multi-correct) |
| 2018 | 0 | — (one limit question filed under this chapter in the archive is not counted) |
| 2017 | 5 | A three-question matching set on f(x) = x + ln x − x ln x: zeros, limits and monotonicity of f, f' and f'' · f'(x) > 2f(x) with f(0) = 1 (multi-correct) · f(x) defined by a determinant in sin x, cos x, sin 2x, cos 2x: critical points and extrema (multi-correct) |
| 2016 | 2 | Least a such that 4ax2 + 1/x ≥ 1 for all x > 0 (single correct) · twice-differentiable f and g with conditions at x = 2: local minimum and second-derivative conclusions (multi-correct) |
| 2015 | 2 | Open cylindrical container of fixed volume: inner radius that minimises material (numerical) · two functions given only through a table of values: which mean-value statements must hold (multi-correct) |
| Total | 15 | 8 multi-correct · 6 single correct (including the 3-question matching set) · 1 numerical |
The Advanced pattern is very different from Main. Advanced rarely asks for a number. It asks you to judge statements about a function: does it have a local minimum, is its derivative increasing, where are its extrema, what must be true given only some conditions. Monotonicity (5 questions) and maxima and minima reasoning (4) lead, followed by critical points and extrema of special functions (3), geometric optimisation (2) and one mean value theorem question. The questions also cluster in 2015–2019. Since 2020 the chapter has given at most one question per year.
The smartest way to decide how to study Application of Derivatives for JEE is to match your time to each sub-topic. Use the time column in your mock tests: if you are consistently slower than the target, that sub-topic needs more practice, not more reading.
| Sub-topic | Main Qs (2024–26) | Advanced Qs (2015–26) | Difficulty | Target time |
|---|---|---|---|---|
| Maxima and minima of a given function | 18 | 4 | Medium | 90 s–2 min |
| Monotonicity | 14 | 5 | Medium–Hard | 90 s–3 min |
| Critical points and counting | 9 | 3 | Medium | 2–3 min (sign chart first) |
| Geometric optimisation | 5 | 2 | Medium–Hard | 3–4 min |
| Rate of change | 1 | 0 | Easy | 60–90 s |
| Rolle's and mean value theorems (Advanced only) | 0 | 1 | Hard | 3–4 min |
Difficulty and target time are our faculty's guidance for exam practice, not exam data.
Maxima and minima of a given function covers local and absolute extreme values, the first and second derivative tests, extrema on closed intervals and questions that ask you to rebuild a polynomial from its extrema. It is the largest block in JEE Main.
Monotonicity asks where a function increases or decreases, or which statements about an abstract function are true. Questions with f'' conditions or functional equations make this the hardest sub-topic in Main.
Critical points and counting asks how many critical points, local maxima or minima, or real roots a function has. One sign chart, drawn carefully, answers most of these.
Geometric optimisation is the classic "largest rectangle" or "shortest distance" problem. The skill is choosing one variable and writing the quantity in terms of it.
Rate of change is a single related-rates idea: differentiate the relation between quantities with respect to time. It appeared once, in 2025.
Rolle's and mean value theorems are no longer in the JEE Main syllabus, but they are still in the JEE Advanced syllabus and appeared there in 2015.
1-on-1 chapter coaching
Two sub-topics gave close to two-thirds of this chapter's JEE Main questions from 2024 to 2026. Work through them with your own faculty, at your pace, with every doubt answered the same day.
Application of Derivatives packs
✔ Free 30-min mentorship call before you pay · ✔ 100% refund guarantee after the first session · ✔ 7-day WhatsApp access to your faculty
Knowing how to study Application of Derivatives for JEE also means knowing what not to do. These are the mistakes we see most often when students solve this chapter's PYQs.
Keep this table open while you practise. Most of the chapter runs on these results, so learning how to study Application of Derivatives for JEE is mostly about applying them quickly and without errors.
| Sub-topic | Formula / result |
|---|---|
| Monotonicity | f' ≥ 0 on an interval (zero only at isolated points) ⇒ f strictly increasing · f' ≤ 0 (zero only at isolated points) ⇒ f strictly decreasing |
| Critical points | c is critical if f'(c) = 0 or f'(c) does not exist (c in the domain of f) |
| First derivative test | f' changes + to − at c ⇒ local maximum · − to + ⇒ local minimum · no change ⇒ no extremum |
| Second derivative test | f'(c) = 0, f''(c) < 0 ⇒ local maximum · f''(c) > 0 ⇒ local minimum · f''(c) = 0 ⇒ test fails, use the first derivative test |
| Higher-order test | If the first non-zero derivative at c is f(n)(c): n even ⇒ extremum (max if f(n)(c) < 0); n odd ⇒ no extremum |
| Absolute extrema | On [a, b], compare f at all critical points and at a and b |
| Integral-defined functions | F(x) = ∫ax f(t) dt ⇒ F'(x) = f(x) · F(x) = ∫u(x)v(x) f(t) dt ⇒ F'(x) = f(v)v' − f(u)u' |
| Polynomial from extrema | Extrema at x = p, q ⇒ f'(x) = k(x − p)(x − q)·(remaining factor) · integrate and use given values |
| Roots of f(x) = 0 | Between two consecutive roots of f' there is at most one root of f · a cubic has 3 distinct real roots ⇔ its local max and local min values have opposite signs |
| Trig extremes | a sin θ + b cos θ ranges over [−√(a2 + b2), √(a2 + b2)] |
| AM–GM check | For positive a, b: a + b ≥ 2√(ab), equality when a = b. Useful to verify a minimum quickly |
| Rate of change | dV/dt = (dV/dr)(dr/dt) · sphere: V = (4/3)πr3, S = 4πr2 · cylinder: V = πr2h |
| Optimisation shapes | Rectangle of fixed perimeter: maximum area when it is a square · closed cylinder of fixed volume: minimum surface area when h = 2r · open cylinder (one base): minimum material when h = r |
| Shortest distance | Shortest distance between two curves lies along their common normal · point to circle: |distance to centre − radius| |
| Advanced: Rolle and LMVT | Rolle: f(a) = f(b) ⇒ f'(c) = 0 for some c in (a, b) · LMVT: f'(c) = [f(b) − f(a)]/(b − a) |
| Advanced: auxiliary functions | f' − kf > 0 ⇒ (e−kx f)' > 0, so e−kx f is increasing |
This is the core of how to study Application of Derivatives for JEE: one sub-topic at a time, starting with the ones that have given the most marks. Do not move on until you can solve the PYQ type listed under each heading without notes.
Every question in this group follows the same four steps:
Worked example in the style of JEE Main 2026: for f(x) = x2025 − x2000 on [0, 1], f'(x) = x1999(2025x25 − 2000). The only critical point inside (0, 1) is where x25 = 2000/2025 = 80/81. f' is negative before it and positive after it, so this is the minimum. The large powers look frightening, but the method has not changed at all.
For "build the polynomial" questions, start from f'(x). If f has extrema at 4 and 5 and is of degree four, f'(x) is a cubic with factors (x − 4)(x − 5) and one more factor given by the other conditions. Integrate, fix the constant from a given value, and then evaluate what is asked.
Trigonometric extremes are a frequent variation. 2026 asked for the least value of an expression quadratic in sin θ and cos θ. Convert it to a single form such as a + b sin 2θ + c cos 2θ, then use the range ±√(b2 + c2).
Practice target: 20 PYQs from this group. For each, write the factorised f'(x) and the sign chart before anything else.
"Factorise the derivative before you do anything else. A derivative written as a product tells you the critical points and the signs at a glance. A derivative left expanded hides both." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
There are two kinds of monotonicity question, and they need different habits.
Functional-equation questions such as 5f(x) + 4f(1/x) = x2 − 2 (JEE Main 2024) need one extra step: replace x by 1/x, solve the two equations for f(x), and only then differentiate.
Practice target: 15 monotonicity PYQs, at least five of them with abstract functions. For the abstract ones, write the derivative of g in terms of f' before reading the options.
Counting questions ask for the number of critical points, local maxima, local minima or real roots. They are almost always solved by one careful sign chart.
Practice target: 10 counting PYQs. For each, draw the number line with every critical point and the sign of f' in each gap.
"Counting questions are not about solving equations. They are about signs. If you can tell me the sign of f' in each gap of the number line, you already know how many maxima and minima there are." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
These are word problems: the largest rectangle inside a region, the largest triangle with vertices on a curve, the shortest distance between two curves, or the container that uses the least material. The method:
For the shortest distance between a parabola and a circle (JEE Main 2025), use the common-normal idea: the shortest distance from the circle is along the line through its centre, so find the point on the parabola nearest to the centre and subtract the radius. JEE Advanced 2015 asked for the radius of an open cylindrical container of fixed volume that needs the least material, which becomes a one-variable minimisation once you use the volume condition to remove the height.
Practice target: 8 optimisation PYQs, including the 2025 rectangle, parallelogram and distance questions. Spend most of your time on the set-up.
This is the smallest group, with one question in three years: a 2025 problem about an ice-cream layer melting off a spherical chocolate ball, asking for a surface area when the layer reaches a given thickness. The method is short:
Practice target: 5 related-rates problems. Do not spend more time than that. The chapter's marks are elsewhere.
That is how to study Application of Derivatives for JEE sub-topic by sub-topic. Once you have finished all five, test yourself under exam conditions with our JEE test series, and bring any question that still does not make sense to the JEE Doubt Lab for a 1-on-1 explanation.
Ready for exam-level practice?
Abstract-function and multi-correct questions are where most students lose marks in this chapter. Practise them 1-on-1 with an IITian Maths HOD who has taught them for years.
Application of Derivatives packs
✔ Free 30-min mentorship call before you pay · ✔ 100% refund guarantee after the first session · ✔ 7-day WhatsApp access to your faculty
If you are deciding how to study Application of Derivatives for JEE Advanced as well as Main, this section is for you. Advanced keeps topics that Main has removed, and it asks familiar topics in a harder form.
8 of the 15 Advanced questions from 2015 to 2026 were "one or more correct". Each option is a separate judgement: "f has a local maximum at 2", "f' is decreasing on (0, 1)", "F(x) ≠ 0 for all x in (0, 5)". Practise by proving or disproving every option, even the ones that look obvious. One wrong tick costs marks.
JEE Advanced 2017 had a three-question matching set on f(x) = x + ln x − x ln x, asking about the zeros, limits and monotonicity of f, f' and f''. The skill is to study one function completely before reading the questions: domain, f', f'', their zeros, their signs and their behaviour as x → 0+ and x → ∞.
These are out of JEE Main but still in JEE Advanced. Advanced 2015 gave two functions only through a table of values and asked which mean-value statements must be true. Advanced 2017 gave f'(x) > 2f(x) with f(0) = 1. The trick there is to notice that (e−2xf(x))' = e−2x(f' − 2f) > 0, so e−2xf(x) is increasing and f(x) > e2x for x > 0.
Advanced 2019 asked about the successive local maxima and minima of f(x) = sin(πx)/x2, and Advanced 2016 asked for the least a such that 4ax2 + 1/x ≥ 1 for all x > 0. The second is really a minimum problem: the inequality holds for all x exactly when the minimum of the left side is at least 1.
None of the 15 Advanced questions from 2015 to 2026 in our count was mainly about tangents and normals, but the topic is still in the JEE Advanced syllabus. Learn the equations of the tangent and normal, the angle between two curves, and the condition for two curves to touch, then move on.
"Advanced does not give you more formulas in this chapter. It gives you less information. When a question gives you only a condition like f' > 2f, look for the function whose derivative is that condition. That one idea solves a surprising number of Advanced questions." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
The last step in how to study Application of Derivatives for JEE is an honest self-check. You should be able to answer "yes" to every line below without notes.
Application of Derivatives gave JEE Main 51 questions from 2024 to 2026 and JEE Advanced 15 questions from 2015 to 2026. In JEE Main, maxima and minima (18) and monotonicity (14) gave nearly two-thirds of the questions, and counting problems added nine more. Tangents and normals gave none, because they are no longer in the JEE Main syllabus. So if you are working out how to study Application of Derivatives for JEE, the order is clear: master the sign chart and the maxima-minima routine first, then monotonicity of explicit and abstract functions, then counting questions, and finally geometric optimisation. If you are also writing JEE Advanced, add Rolle's theorem, LMVT and multi-correct practice on single functions.
How we built this data. We went through every JEE Main shift paper from 2024, 2025 and 2026 and every JEE Advanced paper from 2015 to 2026, picked out each question from this chapter and tagged it by the main idea it tests. Where the archived text of a question was too incomplete to classify with confidence, or a question could not be located, we counted it in the totals and said so instead of guessing. Questions from other chapters that the archive files here are left out of the sub-topic tables, as the footnotes explain.
If you want a faculty member to check your preparation, every pack starts with a free 30-minute mentorship call. See all Maths chapter packs on our JEE Maths coaching page, and once your exam is done, our JEE counselling team can help you turn your rank into the right college and branch.
JEE Main asked 26 questions from this chapter in 2024 (4.33% of Maths questions), 14 in 2025 (2.95%) and 11 in 2026 (2.32%), counting all shifts. That is 51 questions in three years.
JEE Main questions are mostly medium. They follow a fixed routine: differentiate, find critical points, draw a sign chart, compare values. The hardest Main questions give an unknown function through conditions. JEE Advanced questions are harder because most are multi-correct statements about a function. If you follow this guide on how to study Application of Derivatives for JEE, the Main questions become routine with steady practice.
Yes. It was asked 51 times in JEE Main from 2024 to 2026 and 15 times in JEE Advanced from 2015 to 2026. Its methods are also used in Area under Curves, inequalities and root-counting questions in other chapters.
No. It gave 11 to 26 JEE Main questions a year from 2024 to 2026, and the questions repeat the same few types. After the 2024 syllabus change, the JEE Main part of the chapter is smaller, which makes it one of the better returns on study time in Calculus.
Master maxima and minima first, then monotonicity, then counting critical points, extrema and roots. In JEE Main 2024–2026 these three sub-topics gave 41 of the 51 questions, about 80% of the chapter.
JEE Advanced focuses on statements about one function: whether it has local extrema, where it increases, how f, f' and f'' behave. Rolle's theorem and the mean value theorem, which are no longer in JEE Main, are still in the Advanced syllabus. 8 of the 15 Advanced questions from 2015 to 2026 were multi-correct.
