Applications of Derivatives

Concept overview

The derivative f'(x) gives the slope of the tangent to y = f(x) at that x. Tangent line: y − f(a) = f'(a)(x − a). Normal line has slope −1/f'(a). Monotonicity: f is increasing where f' > 0, decreasing where f' < 0, constant where f' = 0 on an interval.

Local extrema occur where f'(x) = 0 (or f' doesn't exist). Second derivative test: f''(x) > 0 at a critical point means local minimum; f''(x) < 0 means local maximum. Global extrema on a closed interval are among critical points and endpoints — evaluate f at each and pick the largest/smallest.

Rolle's theorem: if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) with f'(c) = 0. Mean value theorem generalises: if f is continuous on [a, b] and differentiable on (a, b), there exists c with f'(c) = (f(b) − f(a))/(b − a). MVT is the practical tool — Rolle is a special case.

Rate of change problems apply the chain rule: dV/dt = (dV/dr)(dr/dt). Related-rates problems where multiple quantities change simultaneously are standard JEE fare.

Key formulas & concepts

• Tangent line: y − f(a) = f'(a)(x − a)
• Normal line slope: −1 / f'(a)
• Critical point: f'(x) = 0 or f' undefined
• Second derivative test: f''(x) > 0 → min; f''(x) < 0 → max
• Rolle's theorem: f(a) = f(b) ⇒ ∃ c with f'(c) = 0
• MVT: ∃ c with f'(c) = (f(b) − f(a))/(b − a)
• Increasing/decreasing: f' > 0 (increasing); f' < 0 (decreasing)

Common mistakes to avoid

• Confusing local extrema (interior only) with global extrema (also include endpoints).
• Using second derivative test at points where f'' = 0 — it's inconclusive there; use first-derivative test instead.
• Applying Rolle without checking f(a) = f(b).
• Missing critical points where f'(x) is undefined (cusps, corners).

JEE Main relevance

JEE Main asks 2 questions every paper — usually one tangent/normal problem and one maxima-minima or MVT-application.

JEE Advanced relevance

Advanced problems combine optimisation with geometric constraints (find dimensions maximising volume subject to surface area), or use MVT to prove inequalities.

Previous-year questions (PYQ)

Maxima-minima problems are the top PYQ archetype. Practise 20 in one focused session.