JEE Main Maths · Differentiation PYQ
JEE Main Differentiation Previous Year Questions (2002–2025)
Differentiation and Applications of Derivatives is one of the two highest-weightage Mathematics chapters in JEE Main — ~8% across 10 years with 2 questions per session routinely. Together with Integration and Limits, it forms the Calculus core that decides ~25% of the Maths section. Almost every Maths paper asks a maxima-minima or tangent-normal problem.
Differentiation PYQs from 2002 to 2025, spanning standard derivatives, chain rule, implicit, parametric and logarithmic differentiation, plus applications (maxima-minima, tangent-normal, monotonicity, L'Hôpital for Limits via differentiation). Every solution is KaTeX-rendered with the technique named upfront.
Differentiation & Applications of Derivatives at a Glance
Key Sub-Topics & What's Tested
Standard Derivatives
Derivatives of x^n, e^x, ln x, trigonometric, inverse trigonometric and hyperbolic functions — instant recall required.
Chain Rule & Composition
dy/dx for f(g(x)), nested compositions, differentiation of trigonometric-of-trigonometric expressions.
Implicit Differentiation
Differentiating F(x, y) = 0 with respect to x, solving for dy/dx, higher-order implicit derivatives.
Parametric Differentiation
Given x = f(t), y = g(t), compute dy/dx = (dy/dt) / (dx/dt). Finding second derivative in parametric form.
Logarithmic Differentiation
For y = f(x)^g(x) or products of complex expressions, take ln both sides first. Critical for x^x type problems.
Tangents & Normals
Equation of tangent and normal to curves at a given point, orthogonal curves condition, angle between two curves.
Maxima & Minima
First derivative test, second derivative test, finding extrema in a closed interval, practical word problems (box, cone, triangle).
Monotonicity & Increasing/Decreasing
Intervals where f(x) is increasing/decreasing, points of inflection, using derivatives to compare function values.
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| Direct Derivative Computation | 20% | Find d/dx of ln(sin(e^x)) using chain rule. |
| Implicit / Parametric | 15% | Given x² + y² = 25, find dy/dx at the point (3, 4). |
| Maxima-Minima (Function Extrema) | 25% | Find maximum value of f(x) = x · e^(-x) for x ≥ 0. |
| Tangent / Normal Problems | 15% | Find equation of tangent to y = x² + 3x - 2 at x = 1. |
| Monotonicity / Increasing-Decreasing | 15% | Find the interval where f(x) = x³ - 3x + 2 is increasing. |
| Word Problems (Practical Optimisation) | 10% | Find dimensions of the rectangle of maximum area inscribed in a circle of radius r. |
How to Solve Differentiation & Applications of Derivatives PYQs
- 1Lock the standard derivative table. d/dx of x^n, e^x, ln x, sin x, cos x, tan x, inverse trig functions — instant recall. ~30% of PYQs use these directly.
- 2Chain rule discipline: outside-in. For f(g(h(x))), differentiate f first (treating g as a variable), then multiply by g'(h(x)) · h'(x). Work outwards consistently.
- 3For implicit / parametric, use the formula directly. Parametric dy/dx = (dy/dt) / (dx/dt). Implicit: differentiate both sides with respect to x, isolate dy/dx.
- 4Maxima-minima: always check boundary + critical points. In a closed interval [a, b], extrema can be at x = a, x = b or at points where f'(x) = 0. All three need checking.
- 5For word problems, sketch and define variables first. Don't jump into algebra. Define the quantity to be maximised, set up its formula, then differentiate.
- 6Second derivative test for classification. At a critical point, f''(x) > 0 → local min; f''(x) < 0 → local max; f''(x) = 0 → inconclusive, use first derivative sign change.
Common Mistakes That Cost Marks
- Missing chain rule inside trig arguments. d/dx[sin(x²)] = 2x cos(x²), not cos(x²). The chain rule multiplier is essential.
- Using product rule where log differentiation is cleaner. For y = x^(sin x), logarithmic differentiation handles it in 3 lines; product rule can't handle the variable exponent.
- Forgetting dy/dx in implicit differentiation. When differentiating y² with respect to x, the result is 2y · (dy/dx), not 2y. Missing this factor breaks the equation.
- Second derivative test ambiguity. When f''(x) = 0 at a critical point, the test fails — switch to first derivative sign change method.
- Missing boundary extrema in closed intervals. Maximum of f(x) = x on [0, 1] is at x = 1 (the boundary), not at any f'(x) = 0 point. Always evaluate at endpoints.
Related JEE Main Practice
Frequently asked questions
How many Differentiation PYQs should I solve?
Target 80–100 PYQs across 2010–2025. Given Differentiation shares the #1 weightage slot with Integration (~8% each), over-practising pays off in mock test speed.
Which is harder — Differentiation or Integration PYQs?
Differentiation is mechanically more straightforward (one direction — formulas + chain rule). Integration requires reverse-engineering which technique fits. But maxima-minima word problems in Differentiation can be deceptively tricky — especially practical optimisation.
Do I need to memorise second derivatives of standard functions?
For common functions yes — d²/dx² of sin x = -sin x, cos x = -cos x, e^(ax) = a² e^(ax). These appear in PYQs directly. For others, derive from first derivative as needed.
What's the most-tested maxima-minima pattern?
Rectangle-of-max-area inscribed in a curve (circle, ellipse, parabola), box-of-max-volume given surface constraints, sum-of-cubes minimum. ~5 variants per 10-year window — learn the template.
When should I use L'Hôpital's rule?
Only for 0/0 or ∞/∞ indeterminate forms. Check the indeterminate form first, then apply. L'Hôpital shows up in Limits PYQs — roughly 3–5 direct applications per 10 years.
Do I need calculus beyond single-variable for JEE Main?
No. JEE Main Maths stays within single-variable calculus. Multivariable calculus (partial derivatives, gradients) is not in the syllabus.
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