Chapter 6 — Application of Derivatives

Chapter 6 of the Class 12 Maths NCERT textbook, "Application of Derivatives", teaches how to use derivatives to find rates of change, determine increasing or decreasing intervals of functions, and locate maximum and minimum values using the First and Second Derivative Tests.

• Derivatives as rates of change — The chapter reframes the derivative as a real-world rate — how one quantity changes with another — and uses the chain rule to relate rates through a shared variable, connecting calculus to problems in science, economics and engineering.
• Where functions rise and fall — By reading the sign of f'(x), students learn to map out the intervals where a function increases or decreases, and to pinpoint critical points where the behaviour can turn — the natural candidates for peaks and troughs.
• Finding maxima and minima — Two complementary tests — watching f' change sign and checking the sign of f'' — classify critical points as local maxima, minima or inflexion. A working rule then extends this to closed intervals by comparing values at critical points and endpoints.

1. 01The derivative dy/dx represents the rate of change of y with respect to x; the Chain Rule connects rates of change through an intermediate variable t.
2. 02A function f is increasing on (a, b) if f'(x) > 0 for each x in (a, b), and decreasing if f'(x) < 0 for each x in (a, b).
3. 03A critical point is any point c where f'(c) = 0 or f is not differentiable; critical points are candidates for local maxima or minima.
4. 04First Derivative Test: if f'(x) changes from positive to negative at c, then c is a local maximum; if it changes from negative to positive, c is a local minimum; no sign change means c is a point of inflexion.
5. 05Second Derivative Test: if f'(c) = 0 and f''(c) < 0, then c is a local maximum; if f''(c) > 0, it is a local minimum; if f''(c) = 0, the test fails and the First Derivative Test must be used.
6. 06To find absolute maximum and minimum on a closed interval [a, b], evaluate f at all critical points inside the interval and at the endpoints, then compare all values.