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Class 12 Mathematics Chapter 6 Application of Derivatives — Formulas & Key Points

Q: What is the difference between local maximum and absolute maximum?

A local maximum is the highest point in a small neighbourhood around x = c; there may be other points elsewhere where f is even larger. An absolute maximum is the largest value of f over the entire domain or a specified interval. For example, f(x) = x³ has no absolute maximum on ℝ, but on [−1, 1] the absolute max is f(1) = 1.

Q: When should I use the first-derivative test versus the second-derivative test?

Use the second-derivative test when f''(c) exists and is non-zero—it is faster. If f''(c) = 0 or f''(c) does not exist, fall back on the first-derivative test by checking the sign of f'(x) just to the left and right of c. Both tests are equally valid; choose the one that is easier for the function at hand.

Q: How do I find the point on a curve closest to a given point?

Let the curve be y = f(x) and the given point be (a, b). The distance squared is D² = (x − a)² + (f(x) − b)². Minimising D² is equivalent to minimising D and avoids square roots. Differentiate D² with respect to x, set the derivative to zero, solve for x, then verify it is a minimum using the second-derivative test or checking endpoints if the domain is restricted.

Q: What does it mean when the question says 'find the approximate change in volume'?

This signals the use of differentials: ΔV ≈ dV = (dV/dr) Δr. Compute the derivative dV/dr at the given radius, multiply by the small change Δr, and that is your approximate change. This is faster than computing exact volumes and subtracting, and it is the method CBSE expects when the word 'approximate' appears.

Q: Why must I check endpoints when finding absolute extrema?

Critical points found by setting f'(x) = 0 are only candidates for extrema in the interior of the interval. The absolute maximum or minimum might occur at the boundary points x = a or x = b. For instance, f(x) = x on [0, 1] has no critical points (f'(x) = 1 ≠ 0), but the absolute min is f(0) = 0 and max is f(1) = 1, both at endpoints.

Q: How do I handle a vertical tangent or undefined derivative?

A vertical tangent occurs when f'(x) is undefined (for example, at a cusp or where the denominator of f'(x) is zero). At such a point, the tangent line is x = x₀ (a vertical line), and the normal is horizontal: y = y₀. Always check the definition of f' near the point to confirm the tangent is truly vertical and not just a discontinuity.

Q: What is the fastest way to sketch the sign of f'(x) on a number line?

Factor f'(x) completely and mark the roots on a number line. Choose a test point in each interval between roots and plug it into f'(x) (no need to compute the exact value—just determine + or −). If f'(x) is a product of linear factors, you can use the sign-alternation rule: the sign flips at each simple root but stays the same at a repeated root of even multiplicity.

Q: Can a function have a point of inflection at a critical point?

Yes. A point x = c can simultaneously satisfy f'(c) = 0 (stationary point) and have f''(c) change sign (inflection point). For example, f(x) = x³ has f'(0) = 0 and f''(x) = 6x changes sign at x = 0, so x = 0 is both a stationary point and an inflection point. It is neither a local max nor min because the first-derivative test shows f' does not change sign.

Q: How many marks does Chapter 6 Application of Derivatives carry in the CBSE board exam?

Typically 8–10 marks spread across one or two short-answer questions (2 or 4 marks each) and one long-answer question (6 marks). The exact distribution varies year to year, but derivatives and their applications together (Chapters 5 and 6) form a significant portion of the Calculus unit, which is worth 35 marks in the 80-mark theory paper.

Q: What should I do if f''(c) = 0 in the second-derivative test?

The second-derivative test is inconclusive when f''(c) = 0. Switch to the first-derivative test: examine the sign of f'(x) just before and after c. If f' changes from positive to negative, c is a local maximum; from negative to positive, a local minimum; if f' does not change sign, c is neither. Alternatively, compute higher derivatives f'''(c), f⁽⁴⁾(c), … until you find a non-zero one, though that is rarely needed at Class 12 level.

Chapter 6 Application of Derivatives is one of the highest-scoring sections in the CBSE Class 12 Mathematics board paper, typically carrying 8–10 marks across short and long answers. Mastery of derivatives is not just about differentiation rules—it is about applying those rules to find slopes of tangents, identify intervals where a function rises or falls, locate maxima and minima, and solve real-world rate problems. This formula sheet collects every key formula, definition, and test in one place, structured for last-minute revision and rapid recall during exams.

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Key takeaways

✓The first derivative dy/dx gives the slope of the tangent and reveals whether a function is increasing (f'(x) > 0) or decreasing (f'(x) < 0).
✓The second derivative d²y/dx² determines concavity and classifies critical points as maxima (f''(x) < 0) or minima (f''(x) > 0).
✓Tangent at (x₀, y₀) is y - y₀ = f'(x₀)(x - x₀); the normal is perpendicular with slope -1/f'(x₀).
✓Critical points occur where f'(x) = 0 or f'(x) is undefined; use the first- or second-derivative test to classify them.
✓Absolute extrema on a closed interval require testing critical points and endpoints.
✓Rate of change problems (related rates) require the chain rule: dz/dt = (dz/dx)(dx/dt).
✓Approximations via differentials: Δy ≈ dy = f'(x) dx works when dx is small.

Rate of Change and Related Rates — Core Formulas

The derivative dy/dx represents the instantaneous rate of change of y with respect to x. When quantities are linked through a functional relationship, the chain rule connects their rates. For example, if a ladder slides down a wall, the rates at which the base moves and the height drops are related through the Pythagorean theorem differentiated with respect to time. These problems appear regularly in CBSE board papers as 4-mark or 6-mark questions. Always identify the given rate, the required rate, and the equation connecting the variables before differentiating.

If y = f(x), then dy/dx is the rate of change of y with respect to x.
Related rates: given dz/dt and an equation relating z and x, find dx/dt using the chain rule.
Common scenarios: radius and area of a circle, sides of a triangle, volume and surface area of solids.
Always substitute known values after differentiating, not before.

Increasing and Decreasing Functions — All Definitions and Tests

A function f is strictly increasing on an interval if for any x₁ < x₂ in that interval, f(x₁) < f(x₂). The practical test: f is increasing where f'(x) > 0 and decreasing where f'(x) < 0. To find these intervals, solve f'(x) = 0 to locate critical points, then test the sign of f'(x) in each sub-interval. CBSE board questions often ask students to find intervals of increase or decrease for polynomial or rational functions. Remember to write intervals in proper notation (open or closed brackets) and to exclude points where f'(x) is undefined or f itself is discontinuous.

f is increasing on I if f'(x) ≥ 0 for all x in I (strictly increasing if f'(x) > 0).
f is decreasing on I if f'(x) ≤ 0 for all x in I (strictly decreasing if f'(x) < 0).
Find critical points by solving f'(x) = 0.
Check sign of f'(x) in intervals between critical points using test values.
Exclude points of discontinuity from intervals.

Tangents and Normals — Equation Formulas

The tangent to the curve y = f(x) at the point (x₀, y₀) has slope m = f'(x₀). Its equation in point-slope form is y - y₀ = f'(x₀)(x - x₀). The normal is the line perpendicular to the tangent at that point, so its slope is -1/f'(x₀) (provided f'(x₀) ≠ 0). If the tangent is vertical (f'(x₀) undefined), the normal is horizontal (y = y₀) and vice versa. CBSE board papers regularly set 4-mark questions asking for equations of tangents or normals at given points or where the tangent has a specified slope. Always compute the y-coordinate if only x₀ is given, and simplify your final equation to standard form if required.

Tangent at (x₀, y₀): y - y₀ = f'(x₀)(x - x₀).
Normal at (x₀, y₀): y - y₀ = -1/f'(x₀) (x - x₀).
If f'(x₀) = 0, tangent is horizontal (y = y₀) and normal is vertical (x = x₀).
To find point where tangent is parallel to a line y = mx + c, solve f'(x) = m.
To find point where tangent is perpendicular to a line, solve f'(x) = -1/m.

Maxima and Minima — Critical Points and Classification

A function f has a local maximum at c if f(c) ≥ f(x) for all x near c; a local minimum if f(c) ≤ f(x). Critical points are where f'(x) = 0 or f'(x) does not exist. To classify a critical point x = c, use the first-derivative test (check sign change of f'(x) around c) or the second-derivative test: if f''(c) > 0 then c is a local minimum; if f''(c) < 0 then c is a local maximum; if f''(c) = 0 the test is inconclusive. For absolute extrema on a closed interval [a, b], evaluate f at all critical points in (a, b) and at the endpoints a and b, then compare. The 2024 CBSE board carried a 6-mark question on finding absolute maximum and minimum of a function on a given interval.

Critical points: solve f'(x) = 0 or find where f'(x) is undefined.
First-derivative test: if f' changes from + to − at c, local max; from − to + is local min.
Second-derivative test: f''(c) > 0 ⇒ local min; f''(c) < 0 ⇒ local max.
Absolute extrema on [a, b]: compare f(a), f(b), and f(c) for all critical c in (a, b).
If f''(c) = 0, use the first-derivative test or higher derivatives.

Formula Table — Derivatives, Slopes, and Rates

Below is a consolidated table of every formula you need for Application of Derivatives. Memorise the 'When to Use' column—it tells you the signal words in a question that trigger each formula. For instance, 'find the slope of the tangent' means compute f'(x₀); 'find intervals where f is increasing' means solve f'(x) > 0. Board examiners often test whether students can translate a word problem into the correct mathematical operation. Keep this table on a single sheet during revision and practise writing each formula from memory until it becomes automatic. The 2025 sample papers from CBSE include at least two direct formula-application questions worth 8 marks combined.

Key Terms and Definitions — NCERT Vocabulary

Use the exact NCERT terminology in your board answers to signal to the examiner that you understand the theory. For instance, write 'strictly increasing' rather than 'always going up'; write 'local maximum' not 'peak'. The term 'critical point' means any x where f'(x) = 0 or f'(x) does not exist. 'Stationary point' is a synonym for a point where f'(x) = 0. 'Point of inflection' is where the concavity changes, i.e., f''(x) changes sign; it may or may not be a critical point. The CBSE marking scheme awards full marks for correct terminology and deducts for vague language like 'the function goes up here' instead of 'f is increasing on this interval'.

Critical point: x = c where f'(c) = 0 or f'(c) does not exist.
Stationary point: x = c where f'(c) = 0 (slope of tangent is zero).
Local maximum: f(c) ≥ f(x) for all x near c.
Local minimum: f(c) ≤ f(x) for all x near c.
Absolute (global) maximum: largest value of f on the entire domain or interval.
Absolute (global) minimum: smallest value of f on the entire domain or interval.
Point of inflection: x = c where f''(x) changes sign (concavity changes).
Concave up: f''(x) > 0; graph bends upward like a cup.
Concave down: f''(x) < 0; graph bends downward like a cap.

Memory Tricks and Mnemonics

Remembering whether f''(c) > 0 is a minimum or maximum trips up many students under exam pressure. Mnemonic: 'Positive second derivative = smile = minimum' (a smile curves upward, so the function is concave up and has a valley). For tangent versus normal slopes, recall that their product is −1 (perpendicular lines). When solving related rates, the phrase 'Differentiate the equation with respect to time' is your mantra—write d/dt on both sides and apply the chain rule term by term. Another tip: in word problems about increasing functions, the phrase 'profit is rising' translates to P'(x) > 0. Write these mnemonics on a flashcard and review them the night before your board exam.

f''(c) > 0 ⇒ concave up ⇒ smile ⇒ local minimum.
f''(c) < 0 ⇒ concave down ⇒ frown ⇒ local maximum.
Tangent and normal slopes multiply to −1 (perpendicular).
Related rates: always differentiate the connecting equation with respect to time first, then substitute values.
Increasing ⇔ derivative positive; decreasing ⇔ derivative negative.

Common Mistakes — Sign, Unit, and Notation Pitfalls

Students often write f'(x) > 0 and conclude the function is decreasing—this is a sign error. Remember: positive derivative means increasing. Another frequent mistake is forgetting to check endpoints when finding absolute extrema on a closed interval; the maximum or minimum might occur at x = a or x = b, not at a critical point. In related-rates problems, substituting numerical values before differentiating leads to incorrect answers—always differentiate symbolically first. Units matter: if radius is in cm and time in seconds, then dr/dt has units cm/s and dV/dt has units cm³/s. CBSE examiners deduct marks for missing or incorrect units in word problems. Finally, when the tangent is vertical (f'(x₀) undefined), the normal is horizontal (y = constant), and vice versa—mixing these up costs easy marks.

Sign errors: f'(x) > 0 means increasing, not decreasing.
Forgetting to test endpoints when finding absolute max/min on [a, b].
Substituting values before differentiating in related rates—differentiate first.
Omitting units or using wrong units in applied problems.
Confusing vertical tangent (f' undefined, normal is horizontal) with horizontal tangent (f' = 0, normal is vertical).
Writing f''(c) = 0 and concluding maxima or minima without further test.

Solved Mini-Examples — Step-by-Step Applications

These three worked examples cover the most common question types in CBSE board papers: finding intervals of increase/decrease, writing tangent and normal equations, and locating local extrema. Each solution follows the recommended four-step method: (1) Differentiate. (2) Find critical points. (3) Apply the relevant test. (4) State the answer in proper interval notation or equation form. Practising these templates will build the muscle memory needed to handle any variant on exam day. Notice how each solution explicitly states the theorem or test being used—this mirrors the level of detail CBSE examiners expect in long-answer questions worth 6 marks.

One-Glance Last-Minute Revision Box

Print or screenshot this box and keep it on your desk the morning of the exam. It condenses the entire chapter into ten bullet points. Each point is a trigger: see the word, recall the formula and method. For instance, 'rate of change' should immediately bring dy/dx to mind; 'tangent' should prompt y − y₀ = f'(x₀)(x − x₀). This box replaces flipping through 30 pages of notes. Many CBSE toppers report that a single-page summary like this, reviewed 15 minutes before entering the exam hall, boosts confidence and prevents formula mix-ups. Pair this box with three deep breaths, and you are ready to tackle any Application of Derivatives question the board throws at you.

dy/dx = rate of change of y with respect to x.
f'(x) > 0 ⇒ f increasing; f'(x) < 0 ⇒ f decreasing.
Tangent at (x₀, y₀): y − y₀ = f'(x₀)(x − x₀).
Normal at (x₀, y₀): y − y₀ = −[1/f'(x₀)](x − x₀).
Critical points: f'(c) = 0 or undefined.
Second-derivative test: f''(c) > 0 ⇒ min; f''(c) < 0 ⇒ max.
First-derivative test: sign change of f' around c.
Absolute extrema on [a, b]: check f(a), f(b), and all critical points in (a, b).
Related rates: differentiate equation w.r.t. time, then substitute values.
Approximation: Δy ≈ dy = f'(x) dx for small dx.

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Exam Strategy and Marking Scheme Insights

In the 2024 CBSE Class 12 Mathematics paper, Application of Derivatives appeared in one 2-mark question (find intervals of increase), one 4-mark question (tangent-normal equation), and one 6-mark question (absolute maxima-minima on a closed interval). The marking scheme awards 1 mark for correctly finding f'(x), 1 mark for solving f'(x) = 0, 1 mark for testing intervals or applying the second-derivative test, and remaining marks for the final answer in correct notation. Partial credit is generous if your method is sound even if you make an arithmetic slip, but zero method steps means zero marks even if your final answer is accidentally correct. Always write 'Let f(x) = …' at the start, show the differentiation explicitly, state which test you are using, and box your final answer. These habits align with the CBSE marking rubric and can secure you full marks even under time pressure.

2-mark questions: typically find f'(x) and state increasing/decreasing intervals or write a tangent equation.
4-mark questions: combine two steps—find critical points and classify them, or derive both tangent and normal.
6-mark questions: absolute extrema on [a, b] or applied maxima-minima word problems (geometry or physics context).
Show all working: examiners award step-wise marks; no working = at most 1 mark for correct answer.
Use proper notation: write intervals as (a, b) or [a, b]; equations in standard form; units in word problems.

Frequently asked questions

What is the difference between local maximum and absolute maximum?+

A local maximum is the highest point in a small neighbourhood around x = c; there may be other points elsewhere where f is even larger. An absolute maximum is the largest value of f over the entire domain or a specified interval. For example, f(x) = x³ has no absolute maximum on ℝ, but on [−1, 1] the absolute max is f(1) = 1.

When should I use the first-derivative test versus the second-derivative test?+

Use the second-derivative test when f''(c) exists and is non-zero—it is faster. If f''(c) = 0 or f''(c) does not exist, fall back on the first-derivative test by checking the sign of f'(x) just to the left and right of c. Both tests are equally valid; choose the one that is easier for the function at hand.

How do I find the point on a curve closest to a given point?+

Let the curve be y = f(x) and the given point be (a, b). The distance squared is D² = (x − a)² + (f(x) − b)². Minimising D² is equivalent to minimising D and avoids square roots. Differentiate D² with respect to x, set the derivative to zero, solve for x, then verify it is a minimum using the second-derivative test or checking endpoints if the domain is restricted.

What does it mean when the question says 'find the approximate change in volume'?+

This signals the use of differentials: ΔV ≈ dV = (dV/dr) Δr. Compute the derivative dV/dr at the given radius, multiply by the small change Δr, and that is your approximate change. This is faster than computing exact volumes and subtracting, and it is the method CBSE expects when the word 'approximate' appears.

Why must I check endpoints when finding absolute extrema?+

Critical points found by setting f'(x) = 0 are only candidates for extrema in the interior of the interval. The absolute maximum or minimum might occur at the boundary points x = a or x = b. For instance, f(x) = x on [0, 1] has no critical points (f'(x) = 1 ≠ 0), but the absolute min is f(0) = 0 and max is f(1) = 1, both at endpoints.

How do I handle a vertical tangent or undefined derivative?+

A vertical tangent occurs when f'(x) is undefined (for example, at a cusp or where the denominator of f'(x) is zero). At such a point, the tangent line is x = x₀ (a vertical line), and the normal is horizontal: y = y₀. Always check the definition of f' near the point to confirm the tangent is truly vertical and not just a discontinuity.

What is the fastest way to sketch the sign of f'(x) on a number line?+

Factor f'(x) completely and mark the roots on a number line. Choose a test point in each interval between roots and plug it into f'(x) (no need to compute the exact value—just determine + or −). If f'(x) is a product of linear factors, you can use the sign-alternation rule: the sign flips at each simple root but stays the same at a repeated root of even multiplicity.

Can a function have a point of inflection at a critical point?+

Yes. A point x = c can simultaneously satisfy f'(c) = 0 (stationary point) and have f''(c) change sign (inflection point). For example, f(x) = x³ has f'(0) = 0 and f''(x) = 6x changes sign at x = 0, so x = 0 is both a stationary point and an inflection point. It is neither a local max nor min because the first-derivative test shows f' does not change sign.

How many marks does Chapter 6 Application of Derivatives carry in the CBSE board exam?+

Typically 8–10 marks spread across one or two short-answer questions (2 or 4 marks each) and one long-answer question (6 marks). The exact distribution varies year to year, but derivatives and their applications together (Chapters 5 and 6) form a significant portion of the Calculus unit, which is worth 35 marks in the 80-mark theory paper.

What should I do if f''(c) = 0 in the second-derivative test?+

The second-derivative test is inconclusive when f''(c) = 0. Switch to the first-derivative test: examine the sign of f'(x) just before and after c. If f' changes from positive to negative, c is a local maximum; from negative to positive, a local minimum; if f' does not change sign, c is neither. Alternatively, compute higher derivatives f'''(c), f⁽⁴⁾(c), … until you find a non-zero one, though that is rarely needed at Class 12 level.

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