Independent topic page
AOD full marks: simplify → differentiate → critical points + sign chart (and endpoints on closed intervals) → translate to tangent/normal/increasing language.
01. Tangent: slope f′(x₀), equation y−y₀=f′(x₀)(x−x₀). Normal slope is negative reciprocal when f′≠0.
02. Extrema: solve f′=0 (and undefined points in domain), use first-derivative sign chart or second-derivative test when f′′≠0.
03. On [a,b], global max/min compare critical values with f(a), f(b).
04. Implicit: F(x,y)=0 ⇒ dy/dx=−F_x/F_y (F_y≠0). Compute d²y/dx² only if asked.
05. Advanced lane: extrema on a closed interval are critical points plus endpoints — skip either and global max/min leaks. Open-interval asks need sign-chart behaviour at ends.
06. Tangent/normal geometry: point-slope with dy/dx at the contact point; if implicit, differentiate once and plug (x₀,y₀) before solving for y'.
07. Ugly products/quotients: simplify or take log before differentiating when the ask is only about monotonicity or extrema — chain-rule explosions are time sinks.
Official · 2026
Official syllabus points for Applications of Derivatives only — not the full Calculus portion map. Portion hub: Calculus · Calculus analysis.
No topic-specific exclusions flagged — finish the unit points below for AOD.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusTangents/normals, extrema, and mean-value theorems under Advanced formats.
Syllabus points
2020–2026 paper weight
Topic match inside parent portion Calculus (28.5% of Math tags). Stats below are portion backdrop; matched subtopics show where Applications of Derivatives sits.
Portion-wide type/difficulty mix lives on the Calculus analysis — not repeated here so this page stays on AOD.
01. Dominant subtopics: AOD (18), Definite Integrals (11), Continuity/Differentiability (9), Functions (7).
Cross-topic stems
Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Applications of Derivatives — practise the knowledge bridge, not only the single-chapter drill.
Step 1: solve the two ellipse equations simultaneously for the first-quadrant intersection P. Step 2: differentiate implicitly to get slopes of both tangents at P; convert the angle formula into 4 tan θ. Step 3: for the common area, set up the integral (or standard ellipse-sector form) of the overlapping region and read cot α from that area.
Two different asks on one geometry figure — angle then area — under numerical fatigue.
For two conics, always compute intersection → slopes → area in that fixed order.
Step 1: solve x y' = y − x³ (homogeneous/linear in y/x) to get the family y = Cx − x³/2. Step 2: impose the given condition to fix C, then use y' = 0 to locate the local max. Step 3: compare monotonicity on (1,2) and count positive roots of f = g.
Multi-correct options test DE, calculus and graph reading together; one sign error in the DE kills several options.
After every DE, immediately ask: extrema? monotonicity? intersections with a given g?
Step 1: recover points P, Q, R from the slope/tangent conditions on the parabola. Step 2: find the circumcircle of triangle PQR (perpendicular bisectors). Step 3: read the required radius / relation that also ties to an ellipse option.
Geometry hops across three curve types in one SC item.
Practise ‘points on parabola → circle through them → conic check’ as a single drill.
Step 1: analyse the given limit expression to identify the correct f(x) / derivative behaviour. Step 2: on a related (or following) stem, sketch the region defined by inequalities. Step 3: integrate carefully (logs appear from 1/x-type boundaries).
Back-to-back calculus depth: analytic limit then geometric area.
Pair every limit/AOD drill with one area-of-region numerical the same day.
Step 1: place the ellipse in standard form and introduce eccentric angle or slope form. Step 2: impose the geometric condition (tangent/chord/focus). Step 3: match the simplified relation to the correct option.
Parametric slips create attractive wrong options.
Keep both slope and parametric forms of ellipse tangents fluent.
Step 1: for the ellipse–parabola common-tangent MC, write both tangent conditions and compare. Step 2: later area NUM — find intersection limits first. Step 3: integrate difference of functions; keep trig-equation NUM technique nearby for neighbouring items.
Classic Advanced join: geometry conditions feeding calculus numerics in the same paper.
Any week you revise conics, end with one area-between-curves NUM.
Why methods matter
AOD is translation: calculus → geometry/language. These methods make sign charts and point-slope automatic so extrema and tangents stop feeling ‘graphical and vague’.
Cuts chain-rule explosions that steal time and create sign errors.
Do thisRewrite logs/products before chain-rule explosions.
Cuts algebra time and errors.
ln((1+x)/(1−x))=ln(1+x)−ln(1−x) before differentiating.
Extrema become a table, not a vague graph story.
Do thisCritical points → test f′ on either side.
Reliable when f′′=0 or messy.
f=x³−3x → f′=3(x−1)(x+1) → local max −1, local min 1.
Global max/min on [a,b] stop leaking marks at the boundary.
Do thisOn [a,b], always compare f(a), f(b) with critical values.
Global extrema often sit at endpoints.
A local max inside may not be the global max on a closed interval.
Tangent/normal become mechanical — that is how they should feel.
Do thisDerivative gives slope; geometry finishes with point-slope form.
Keeps tangent/normal mechanical.
y=x² at x=1: slope 2, tangent y−1=2(x−1).
Processed examples
Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.
Equation of tangent to y=x² at x=1.
Point-slope is the entire tangent method — no first principles in the hall.
Model chosenm=f′(1)=2; point (1,1).
Answery = 2x − 1
Takeaway: Derivative → point-slope → done.
Find local maxima/minima of f(x)=x³−3x.
Sign chart replaces vague cubic-shape memory.
Model chosenf′=0 → sign chart.
AnswerLocal max (−1,2); local min (1,−2)
Takeaway: Critical points + signs; endpoints if closed interval.
For a > 0, stationary points of f(x) = x³ − 3ax are 4 units apart. Find a and the vertical gap between the local extrema.
Geometry of critical points becomes an equation for the parameter — classic AOD twist.
Model chosenf′ = 0 → separation 2√a = 4 → evaluate extrema.
Answera = 4; vertical gap = 32
Takeaway: Translate geometric facts about extrema into equations on critical points.
Reusable models
Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.
Looks likeEquation of tangent or normal
Fast modelPoint + m=f′(x₀) → point-slope; normal uses −1/m.
Pure template — no re-deriving from first principles.
Do not differentiate from first principles in the exam.
Looks likeIncreasing intervals, local/global extrema
Fast modelf′=0 critical points → sign chart; on [a,b] also compare endpoints.
Sign chart beats vague graphing under time pressure.
Do not skip endpoints on closed-interval global asks.
Looks likeF(x,y)=0; dy/dx or tangent to implicit curve
Fast modeldy/dx=−F_x/F_y; only compute second derivative if asked.
One formula replaces solving for y explicitly.
Do not solve the implicit relation for y unless easy.
Looks likedy/dt from dx/dt; linear approximation / differentials
Fast modelChain rule: dy/dt = y′(x) dx/dt; Δy ≈ f′(x)Δx for approximation.
One chain-rule line ends most rate stems.
Do not treat rates as static slopes without time.
Hall timing
Note domain (closed interval?). Simplify expression before differentiating.
Include points where f′ undefined if they are in domain (corners/cusps).
On closed interval, evaluate endpoints for global extrema.
Wording: increasing vs strictly increasing — match the option language.
After the model is locked
y−y₀ = f′(x₀)(x−x₀)
Tangent at a point
slope = −1/f′(x₀) if f′(x₀)≠0
Normal line
(uv)′=u′v+uv′; (u/v)′=(u′v−uv′)/v²; (f∘g)′=f′(g)g′
All routine derivatives
dy/dx = −F_x/F_y
F(x,y)=0
f′ changes +→− local max; −→+ local min
Sign chart extrema
f′(c)=0, f′′(c)>0 local min; f′′(c)<0 local max
When f′′ easy and nonzero
f′>0 increasing; f′<0 decreasing (on interval)
Increase/decrease intervals
Make it automatic