Class 12 Maths · Chapter 6
Unit: Calculus
Tentative Study Share: 7.0 Marks
Chapter 6: Applications Of Derivatives — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Applications Of Derivatives.
NCERT Exercises in Chapter 6: Applications Of Derivatives
📘 NCERT Concept Notes — Applications Of Derivatives
01 · Rates need a variable and a unit
A derivative with respect to radius is not a time rate. When quantities change together, write their geometric relationship first and differentiate with respect to time. Keep increasing and decreasing signs.
02 · Use a sign chart, not one test point
Derivative signs describe behaviour on intervals. Isolated derivative zeros need not split an increasing interval. A zero derivative at one point does not prove an extremum.
03 · Local tests and absolute extrema
First derivative sign changes classify local extrema. A second derivative can settle a stationary point if it is nonzero. For a closed interval, include both endpoints when comparing values.
04 · Optimisation starts with a feasible domain
Express the quantity to optimise in one variable using the constraint. Reject impossible stationary points and check endpoints or limiting behaviour. Geometry determines the domain before calculus begins.
⚡ Key Formulas & Results — Applications Of Derivatives
Related rates
How do you link a time rate to a derivative with respect to x?
\(\frac{dy}{dt}=\frac{dy}{dx}\frac{dx}{dt}\)
Derive the spherical volume rate
What is the radius factor in dV/dt for a sphere?
\(\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\)
Increasing and decreasing
Which derivative signs prove strict monotonicity?
\(f\prime>0\Rightarrow f\text{ strictly increasing},\quad f\prime<0\Rightarrow f\text{ strictly decreasing}\)
Critical points
Which points are candidates for local extrema?
\(f\prime(c)=0\quad\text{or}\quad f\prime(c)\text{ does not exist}\)
First derivative test
Which sign changes give a local maximum or minimum?
\(+\to-:\text{ maximum};\qquad-\to+:\text{ minimum}\)
Second derivative test
When does f″ classify a stationary point?
\(f\prime(c)=0:\quad f^{\prime\prime}(c)>0\Rightarrow\min,\quad f^{\prime\prime}(c)<0\Rightarrow\max\)
Absolute extrema on a closed interval
Which values must be compared?
\(\{f(a),f(b),f(c_1),\ldots,f(c_k)\}\)
Derive the maximum-volume open box
For the 18 cm square sheet, which corner cut maximises volume?
\(V=x(18-2x)^2,\quad x=3\text{ cm},\quad V_{max}=432\text{ cm}^3\)
🎯 CBSE 2026 Board PYQs from Applications Of Derivatives
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q7
The least value of f(x) = x³ − 12x, x ∈ [0, 3] is
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q22
A room freshener bottle in the shape of an inverted cone sprays the perfume at regular intervals such…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q36
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹300…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q18
Find the absolute maximum of x²+1 on [−5,2].
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q38
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹300…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q11
Find the minimum of e raised to −x on [0,3].
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q37
An online delivery company in a city has 5000 subscribers and collects annual subscription fees of ₹300…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q8
Which statement holds for f(x)=x+1/x, x≠0?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q23
Find the absolute maximum of cos x+sin²x on [0,π]. OR The volume of a solid hemisphere increases at a…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q33
For f(x)=arccot(sin x+cos x), x∈(0,π), find its increasing and decreasing intervals. OR A rectangle of…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q2
Which statement holds for f(x)=x+1/x, x≠0?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q21
Find the absolute maximum of cos x+sin²x on [0,π]. OR The volume of a solid hemisphere increases at a…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q35
Find the intervals of increase and decrease for arctan(sin x−cos x) on (0,π). OR Revolve a rectangle of…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q13
Which statement holds for f(x)=x+1/x, x≠0?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q24
Find the absolute maximum of cos x+sin²x on [0,π]. OR The volume of a solid hemisphere increases at a…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q32
Find the intervals of increase and decrease for ln(sin x+cos x) on (0,π/2). OR Revolve a rectangle of…
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q9
Find the rate of change of a sphere’s volume with respect to diameter when its radius is 5 cm.
CBSE Class 12 Mathematics 2026 Main Paper 65/3/1 · Q22
Find where f(x)=tan x−4x increases on (0,π/2).
📝 Chapterwise Practice Questions — Applications Of Derivatives (109 Questions)
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximal open interval on which f(x)=x²e^(−x) has positive derivative.
Class 12 Applications Of Derivatives · Mcq Single Correct
For f(x)=x²−4x+6, identify the maximal open interval where f′(x)>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
For x>0, at which x is f(x)=x^x stationary?
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum value of (1/x)^x for x>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximum of [x(x−1)+1]^(1/3) on 0≤x≤1.
Class 12 Applications Of Derivatives · Mcq Single Correct
A trapezium area model is A(x)=(10+x)√(100−x²) cm² for 0≤x≤10. Find its maximum.
Class 12 Applications Of Derivatives · Mcq Single Correct
A 5 m ladder rests against a vertical wall. Its top descends at 10 cm/s. Find the magnitude of the rate…
Class 12 Applications Of Derivatives · Mcq Single Correct
Find dA/dr for a circle when r=6 cm.
Class 12 Applications Of Derivatives · Mcq Single Correct
Revenue is R(x)=3x²+36x+5 rupees. Find marginal revenue at x=15.
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the minimum of x²−8x+17 over real x.
Class 12 Applications Of Derivatives · Mcq Single Correct
For f(x)=2x³−15x²+36x+6, identify all maximal open intervals on which f′(x)>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
A right circular cone of height h is inscribed in a sphere of radius r, with 0<h<2r. Which expression…
Class 12 Applications Of Derivatives · Mcq Single Correct
For f(x)=x³+6x²+6, identify the maximal open intervals where f′(x)>0.
Class 12 Applications Of Derivatives · Mcq Single Correct
At which real x does x−x² attain its maximum?
Class 12 Applications Of Derivatives · Mcq Single Correct
A wire of length 20 cm forms the two radii and arc of a circular sector. Find the maximum enclosed area.
Class 12 Applications Of Derivatives · Mcq Single Correct
For a>0,b>0,x>0, find the least value of ax+b/x.
Class 12 Applications Of Derivatives · Mcq Single Correct
Which statement about f(x)=2x³−3x²−36x+7 is correct?
Class 12 Applications Of Derivatives · Mcq Single Correct
For f(x)=2sin³x−3sin²x+12sin x+5 on [0,π/2], which statement holds?
Class 12 Applications Of Derivatives · Mcq Single Correct
For which real b is f(x)=x+cos x+b strictly decreasing on R?
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the global minimum value of f(x)=x⁴−x²−2x+6.
Class 12 Applications Of Derivatives · Mcq Single Correct
Find the maximal open interval where f(x)=2x³+9x²+12x−1 is strictly decreasing.
Class 12 Applications Of Derivatives · Mcq Single Correct
For which values of a is f(x)=a(x−cos x) strictly decreasing on R?
Class 12 Applications Of Derivatives · Assertion Reason
Assertion: For a circle, dA/dr at r=6 cm equals 12π cm²/cm. Reason: The rate of change of circle area…
Class 12 Applications Of Derivatives · Assertion Reason
Assertion: The function f(x)=tan x−x is strictly increasing on each interval (−π/2+kπ,π/2+kπ), k∈Z…