Application of Derivatives

2.1 Core concepts

• The derivative dy/dx (or f'(x)) is interpreted as the rate of change of y with respect to x, and (dy/dx) at x = x0 is the rate of change at that specific point (NCERT §6.2, p. 147).
• If x = f(t) and y = g(t) both vary with a third variable t, then by Chain Rule dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0 — this is the bridge used for every "related rates" problem (NCERT §6.2, p. 147).
• dy/dx is positive if y increases as x increases and negative if y decreases as x increases (NCERT §6.2, Note p. 149).
• Marginal cost = dC/dx and marginal revenue = dR/dx are instantaneous rates of change at the current level of output, illustrated with C(x) = 0.005x³ – 0.02x² + 30x + 5000 and R(x) = 3x² + 36x + 5 (NCERT §6.2, Examples 5–6, p. 150).
• A function f is increasing on I if x1 < x2 ⇒ f(x1) < f(x2) (decreasing reverses the inequality); strict versions use strict inequalities, and "constant" means f(x) = c on I (NCERT §6.3, Definition 1, p. 152).
• First-derivative test for intervals: if f is continuous on [a, b] and differentiable on (a, b), then f is increasing on [a, b] when f'(x) > 0 on (a, b), decreasing when f'(x) < 0, and constant when f'(x) = 0 (NCERT §6.3, Theorem 1, p. 153).
• A function is increasing/decreasing at a point x0 if there exists an open interval around x0 on which it is increasing/decreasing (NCERT §6.3, Definition 2, p. 153).
• To find intervals of increase/decrease: differentiate, solve f'(x) = 0 to get critical points, partition the real line, and test the sign of f' in each subinterval — illustrated for x² – 4x + 6 (decreasing on (–∞, 2), increasing on (2, ∞)) and 4x³ – 6x² – 72x + 30 (NCERT §6.3, Examples 10–11, pp. 155–156).
• For maxima/minima on an interval: f has a maximum at c if f(c) ≥ f(x) for all x ∈ I, and a minimum at c if f(c) ≤ f(x) for all x ∈ I (NCERT §6.4, Definition 3, p. 160).
• A point c is a critical point of f if either f'(c) = 0 or f is not differentiable at c (NCERT §6.4, Note p. 164).
• Theorem 2: If f has a local maximum or minimum at c, then either f'(c) = 0 or f is not differentiable at c. The converse fails — f(x) = x³ has f'(0) = 0 but 0 is a point of inflection, not an extremum (NCERT §6.4, p. 164).
• First Derivative Test (Theorem 3): at a critical point c, if f'(x) changes from + to – as x increases through c, c is a local maximum; from – to +, local minimum; no sign change, point of inflection (NCERT §6.4, p. 164).
• Second Derivative Test (Theorem 4): if f'(c) = 0 and f''(c) < 0, c is a local maximum; if f'(c) = 0 and f''(c) > 0, c is a local minimum; if both are zero, the test fails and one returns to the first derivative test (NCERT §6.4, p. 166).
• Every continuous function on a closed interval [a, b] attains its absolute maximum and absolute minimum at least once on that interval (NCERT §6.4.1, Theorem 5, p. 172).
• Working rule for absolute extrema on [a, b]: (1) find all critical points in (a, b), (2) include the endpoints a and b, (3) evaluate f at all these points, (4) the largest value is the absolute maximum and the smallest is the absolute minimum (NCERT §6.4.1, p. 172).
• Monotonic (purely increasing or purely decreasing) functions attain their maximum and minimum at the endpoints of their domain (NCERT §6.4, p. 162).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig 6.1, p. 152: parabola y = x², showing graph decreases left of origin and increases right of origin — the visual anchor for "increasing/decreasing".
• Fig 6.2, p. 153: the three archetypes — strictly increasing, strictly decreasing, and "neither increasing nor decreasing".
• Fig 6.3, p. 155: sign chart for f(x) = x² – 4x + 6 split at x = 2.
• Fig 6.4, p. 155 and the table on p. 156: sign chart for 4x³ – 6x² – 72x + 30 across (–∞, –2), (–2, 3), (3, ∞).
• Fig 6.7, p. 161: graphical illustration of maximum and minimum at a point (including a non-differentiable case).
• Fig 6.11, p. 163: turning points A, B, C, D on a graph — A and C are valleys (local minima), B and D are hills (local maxima).
• Fig 6.12 and Fig 6.13, pp. 163–164: local maximum/minimum geometry and the counter-example y = x³ at x = 0 (point of inflection).
• Fig 6.19, p. 172: continuous function on a closed interval [a, d] with absolute maximum at the endpoint a and absolute minimum at the endpoint d — used to motivate the closed-interval working rule.

2.5 Key formulas & theorems

2.6 Solved examples (NCERT-grounded)

Example A (NCERT Example 2, p. 148). Cube volume V = x³ increases at 9 cm³/s. Find dS/dt when x = 10.

Step 1 — relate dV/dt to dx/dt: dV/dt = 3x²·dx/dt ⇒ dx/dt = 3/x². Step 2 — surface S = 6x², so dS/dt = 12x·dx/dt: = 12x·(3/x²) = 36/x. Step 3 — at x = 10: dS/dt = 36/10 = 3.6 cm²/s.

Example B (NCERT Example 3, p. 149). Stone-and-wave: dr/dt = 4 cm/s; find dA/dt at r = 10.

Step 1 — A = πr²: dA/dt = 2πr·dr/dt. Step 2 — substitute: 2π(10)(4) = 80π. Step 3 — answer: 80π cm²/s.

Example C (NCERT Example 10, p. 155). Find intervals where f(x) = x² − 4x + 6 is increasing/decreasing.

Step 1 — derivative: f'(x) = 2x − 4. Step 2 — critical point: f'(x) = 0 ⇒ x = 2. Step 3 — sign analysis: f' < 0 for x < 2 (decreasing); f' > 0 for x > 2 (increasing). Decreasing on (−∞, 2), increasing on (2, ∞).

Example D (NCERT Example 20, p. 167). Local extrema of f(x) = 3x⁴ + 4x³ − 12x² + 12.

Step 1 — f': 12x(x − 1)(x + 2) = 0 at x = 0, 1, −2. Step 2 — f'': 36x² + 24x − 24. Evaluate: f''(0) = −24, f''(1) = 36, f''(−2) = 72. Step 3 — classify: x = 0 local max; x = 1 and x = −2 are local minima.

Example E (NCERT Example 27, p. 173). Absolute max/min of f(x) = 2x³ − 15x² + 36x + 1 on [1, 5].

Step 1 — f'(x) = 6(x−2)(x−3): critical points x = 2, 3 (both in [1, 5]). Step 2 — evaluate f at 1, 2, 3, 5: 24, 29, 28, 56. Step 3 — pick extremes: abs max = 56 at x = 5; abs min = 24 at x = 1.

2.4 Common confusions / NTA trap points

• f'(c) = 0 ⇏ extremum. Students assume every stationary point is a max or min; x³ at x = 0 is the textbook counter-example. Always check the sign change of f' (or use the second derivative test).
• "Increasing" vs "strictly increasing". NCERT distinguishes f(x1) ≤ f(x2) (increasing) from f(x1) < f(x2) (strictly increasing). NTA often slips one for the other in distractors.
• Local vs absolute extrema. The absolute extremum on a closed interval can occur at an endpoint, where f' need not vanish. Students who only check f'(x) = 0 miss it (see Theorem 6 and the working rule, p. 172).
• Sign of dV/dt vs dV/dr. "Volume is increasing at 9 cm³/s" gives dV/dt, not dV/dr. NTA distractors invert this regularly.
• Second derivative test fails when f''(c) = 0. Students mistakenly conclude "no extremum" — the correct response is to fall back to the first-derivative test (NCERT §6.4, Theorem 4, p. 166).