Schooling · Class 12 · Mathematics · Application of Derivatives · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-I
Application of Derivatives: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
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Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Rate of change
- Increasing and decreasing functions
- Tangent and normal
- Approximation using differentials
- Local maxima and minima
- First derivative test
- Second derivative test
- Formulas and facts
- Rate of change of y w.r.t. x = dy/dx
- Slope of tangent at (a, f(a)) = f′(a)
- Formula / Rule
- Approximate change: δy ≈ f′(x) δx
- Worked examples
- Rate of change
- Equation of tangent and normal
- Finding maximum and minimum
- Watch out for
- Forgetting to check endpoints when…
- Using the second derivative test…
- Writing the normal slope as f′(a)…
- Mixing up increasing/decreasing with…
- Ignoring units in rate-of-change…
What this chapter is about
This chapter shows how the derivative, which you learned to compute in the previous chapter, becomes a powerful tool for analysing functions and solving practical problems. You move from the question "how do I find dy/dx?" to "what can dy/dx tell me about the behaviour of y?"
Key ideas
- Rate of change: If a quantity y depends on x, then dy/dx gives the instantaneous rate of change of y with respect to x. When x is time, this becomes velocity (for displacement) or the rate of growth (for population, cost, etc.).
- Increasing and decreasing functions: On an interval where dy/dx > 0 the function rises as x increases; where dy/dx < 0 the function falls. At points where dy/dx = 0 or is undefined, the function may change its trend.
- Tangent and normal: At a point (a, f(a)) on the curve y = f(x), the tangent has slope f′(a) and the normal has slope −1/f′(a) (provided f′(a) ≠ 0). The tangent touches the curve; the normal is perpendicular to it.
- Approximation using differentials: For a small change δx in x, the change in y is approximately dy = f′(x) × δx. This lets you estimate values like √4.02 without a calculator.
- Local maxima and minima: A function has a local maximum at c if f(c) ≥ f(x) for all x near c, and a local minimum if f(c) ≤ f(x) for all x near c. These occur where f′(c) = 0 or f′(c) does not exist.
- First derivative test: If f′ 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; if it does not change sign, c is a point of inflection.
- Second derivative test: At a critical point where f′(c) = 0, if f″(c) < 0 then c is a local maximum; if f″(c) > 0 then c is a local minimum; if f″(c) = 0 the test is inconclusive.
- Absolute extrema on a closed interval: Evaluate f at every critical point inside [a, b] and at the endpoints a and b; the largest value is the absolute maximum, the smallest is the absolute minimum.
Formulas and facts to remember
- Formula / Rule: Rate of change of y w.r.t. x = dy/dx · Meaning: How fast y changes per unit change in x
- Formula / Rule: Slope of tangent at (a, f(a)) = f′(a) · Meaning: Gradient of the line that just touches the curve there
- Formula / Rule: Equation of tangent: y − f(a) = f′(a)(x − a) · Meaning: Point-slope form using the derivative
- Formula / Rule: Slope of normal = −1/f′(a) (when f′(a) ≠ 0) · Meaning: Perpendicular slope
- Formula / Rule: Approximate change: δy ≈ f′(x) δx · Meaning: Linear approximation for small δx
- Formula / Rule: Critical point: f′(c) = 0 or f′(c) undefined · Meaning: Candidates for local extrema
- Formula / Rule: First derivative test: sign change of f′ around c decides max/min · Meaning: Uses behaviour on both sides of c
- Formula / Rule: Second derivative test: f″(c) < 0 ⇒ max; f″(c) > 0 ⇒ min · Meaning: Concavity decides the type
Common mistakes
- Forgetting to check endpoints when finding absolute extrema on a closed interval → always evaluate f at both endpoints and at every interior critical point.
- Using the second derivative test when f″(c) = 0 and wrongly concluding there is no extremum → when f″(c) = 0 the test fails; apply the first derivative test instead.
- Writing the normal slope as f′(a) instead of −1/f′(a) → remember the normal is perpendicular, so multiply the tangent slope by −1 and take the reciprocal.
- Mixing up increasing/decreasing with concavity → f′ > 0 means increasing; f″ > 0 means concave upward; these are different properties.
- Ignoring units in rate-of-change problems → carry units through (e.g., m/s, m²/s) to catch dimensional errors.
Quick revision
- dy/dx > 0 ⇒ function increasing; dy/dx < 0 ⇒ function decreasing.
- Tangent slope = f′(a); normal slope = −1/f′(a).
- Critical points occur where f′ = 0 or f′ does not exist.
- First derivative test: check sign change of f′ across the critical point.
- Second derivative test: f″ < 0 gives max, f″ > 0 gives min, f″ = 0 is inconclusive.
- For absolute extrema on [a, b], compare f at critical points and at a and b.