Rates of change, increasing and decreasing functions, and maxima and minima (first and second derivative tests, absolute extreme values on a closed interval, and optimisation word problems). Our own step-by-step solutions to Exercises 6.1 to 6.3 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. \(\tfrac{dy}{dx}\) is the rate of change of y with respect to x. When two quantities change with time, differentiate the relation between them with respect to t (chain rule: \(\tfrac{dA}{dt} = \tfrac{dA}{dr} \cdot \tfrac{dr}{dt}\)) and substitute the instant's values only after differentiating. Marginal cost / revenue is the derivative of total cost / revenue.
What it tests. On an interval, \(f'(x) > 0\) gives increasing and \(f'(x) < 0\) gives decreasing. Find where \(f'(x) = 0\) (or is undefined), split the line at those points, and test the sign of \(f'\) on each piece.
What it tests. Critical points: \(f'(c) = 0\) (or f' undefined). First derivative test: \(f'\) changes + to − gives a local maximum, − to + a local minimum, no change neither. Second derivative test: \(f''(c) < 0\) max, \(> 0\) min. On a closed interval the absolute extremes are among the critical points and the end points. In word problems, write the quantity as a function of one variable first.
What it tests. Mixed practice: set up the function to optimise from the geometry, find critical points, confirm max or min with a derivative test, and read the question for what is asked (a dimension, a value, or an angle). In Q16 the numbers are built on \(\pi = 3.14\) (314 = 100 × 3.14), so take \(\pi = 3.14\) there, as the options do.
Application of Derivatives has 40 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Application of Derivatives · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.