💯 100 Super Revision Test – CBSE Class 12 Applications of Derivatives (2026)
Topic: Applications of Derivatives (AOD) — Tangents, Normals, Monotonicity, Maxima–Minima, Approximation, Rate of Change.
Type: Concept + Derivative-based Objective Practice.
For: CBSE 2026 Aspirants.
Mathematics is not just about solving numbers — it’s about understanding the logic that drives every concept. The chapter “Applications of Derivatives” in CBSE Class 12 Maths is one of the most important and scoring parts of the syllabus. Whether it’s tangents and normals, increasing and decreasing functions, maxima and minima, or rate of change, this topic forms the backbone of calculus-based questions in boards and competitive exams like JEE and CUET.
To help you master every concept through practice, we’ve created a “100 Super Revision Test” — a complete set of carefully designed question–answer pairs that blend conceptual understanding with exam-style problem-solving.
Each question has been selected to:
✅ Strengthen your fundamentals
✅ Cover all NCERT and beyond-level problems
✅ Boost your confidence for the 2026 CBSE Board Exam
So, get your notebook ready, sharpen your problem-solving skills, and start revising the Applications of Derivatives like never before! 🚀
🔹 Structure Example:
- Section 1: Basic Concept Questions
- Section 2: Tangents and Normals
- Section 3: Rate of Change
- Section 4: Increasing & Decreasing Functions
- Section 5: Maxima and Minima
Let’s begin your ultimate revision journey! 💪
- Find the slope of the tangent to \( y = x^2 + 3x + 2 \) at \( x = 1 \).
- Find the equation of tangent to \( y = x^3 \) at \( x = 2 \).
- Find the normal to \( y = x^2 \) at \( x = 3 \).
- At what point on \( y = x^2 - 5x + 6 \) is the tangent parallel to the line \( y = 2x + 1 \)?
- If \( y = \sin x \), find \( \frac{dy}{dx} \) at \( x = \pi/4 \).
- Find the equation of tangent to \( y = e^x \) at \( x = 0 \).
- If \( y = \log x \), find slope of tangent at \( x = e \).
- Find the slope of the curve \( y = \tan x \) at \( x = \pi/4 \).
- Find \( x \) if the tangent to \( y = x^2 \) is perpendicular to line \( y = -x \).
- If \( y = \sqrt{x} \), find the normal at \( x = 4 \).
- Find the intervals of monotonicity for \( f(x) = x^3 - 6x^2 + 9x \).
- Determine the increasing/decreasing nature of \( f(x) = \sin 2x \) on \( [0, \pi] \).
- Find the point where \( f(x) = x^3 - 3x^2 + 4 \) has a local minimum.
- Find the local maxima of \( f(x) = x^3 - 3x + 2 \).
- Find the greatest and least value of \( f(x) = 3x^2 - 6x + 2 \).
- Find the coordinates of turning points of \( f(x) = x^4 - 4x^3 + 6x^2 + 1 \).
- Find the point of inflection of \( f(x) = x^3 - 6x^2 + 9x \).
- Find maximum and minimum values of \( f(x) = 4\sin x + 3\cos x \).
- Find the value of \( x \) where \( f(x) = \tan x \) is increasing fastest.
- Find the point where tangent to \( y = \ln x \) makes an angle of \( 45^\circ \) with the x-axis.
- The radius of a circle increases at rate \( 3\,cm/s \). Find rate of change of area when radius = 7 cm.
- The side of a cube increases at 2 cm/s. Find the rate of change of volume when side = 5 cm.
- A balloon rises vertically at 5 m/s. How fast is its shadow moving on ground when height = 10 m and sun’s elevation is \( 30^\circ \)?
- Find \( \frac{dy}{dx} \) if \( y = x^x \).
- Find rate of change of \( y = \sin^{-1} x \) at \( x = 1/2 \).
- Find the approximate change in \( y = \sqrt{x} \) when \( x \) changes from 25 to 25.5.
- Find approximate value of \( \sqrt{100.4} \).
- Find approximate value of \( (2.01)^5 \) using differentiation.
- Find approximate change in \( \tan x \) when \( x \) increases from \( \pi/4 \) to \( \pi/3 \).
- If \( y = x^3 - 2x + 4 \), find \( \frac{dy}{dx} \) when \( x = 2 \).
- For \( y = \frac{x}{x+1} \), find slope at \( x = 1 \).
- If \( y = \sin x \cos x \), find \( \frac{dy}{dx} \).
- Find derivative of \( e^{2x} \sin x \).
- If \( y = \ln(\sin x) \), find \( \frac{dy}{dx} \).
- If \( y = x^x \), find \( \frac{d^2y}{dx^2} \) at \( x = 1 \).
- Find the maximum of \( f(x) = 2x^3 - 9x^2 + 12x + 5 \).
- Find minimum value of \( f(x) = x + \frac{1}{x} \) for \( x > 0 \).
- Find maximum of \( f(x) = x(10 - x)^2 \).
- Find \( k \) if \( y = x^2 + kx + 5 \) has a minimum value 2.
- Find points of local extrema of \( f(x) = x^4 - 4x^3 \).
- Show that \( y = \sin x \) is increasing on \( [0, \pi/2] \).
- Show that \( y = x^3 \) is strictly increasing for all real x.
- Show that \( y = x^2 \) is decreasing on \( (-\infty, 0] \).
- Find the intervals where \( f(x) = \cos 2x \) is decreasing.
- For \( f(x) = |x| \), discuss differentiability at 0.
- Find slope of tangent to \( y = |x-2| \) at \( x = 3 \).
- Find the equation of tangent to \( y = \sin^{-1} x \) at \( x = 1/2 \).
- Find tangent and normal at \( x = 1 \) for \( y = \log(1+x) \).
- If \( y = a^x \), prove \( \frac{dy}{dx} = a^x \ln a \).
- If \( y = \tan^{-1} x \), find \( \frac{dy}{dx} \).
- Find \( \frac{d}{dx}(\sin^{-1}(2x)) \).
- Find \( \frac{d}{dx}(\cos^{-1}(\sqrt{x})) \).
- Find \( \frac{d}{dx}(\tan^{-1}(3x)) \).
- Find slope of \( y = \tan^{-1}(x^2) \) at \( x = 1 \).
- Find maxima/minima of \( f(x) = x^3 - 3x^2 + 4x + 7 \).
- Find \( x \) where \( f(x) = 2x^2 - 8x + 5 \) is minimum.
- Find the greatest and least value of \( f(x) = \sin x + \cos x \).
- Find rate of change of volume of sphere when radius = 7 cm, \( \frac{dr}{dt} = 2 \,cm/s \).
- Find the instantaneous rate of change of \( y = 3x^2 + 2x \) at \( x = 4 \).
- Find the equation of tangent to \( y = \sqrt{x} \) at \( x = 9 \).
- If \( y = \frac{x^2 + 1}{x} \), find \( \frac{dy}{dx} \) at \( x = 2 \).
- If \( y = x^3 - 3x + 1 \), find \( \frac{dy}{dx} \) and point of maxima.
- Find where \( f(x) = \sin 2x - \cos x \) attains its maxima.
- Find slope of tangent to \( y = e^{-x} \) at \( x = 0 \).
- Find rate of change of perimeter of circle when radius = 10 cm, \( \frac{dr}{dt} = 0.5\,cm/s \).
- Find \( x \) where \( f(x) = 3x^4 - 4x^3 \) has a turning point.
- Find the normal to \( y = x^2 + 4x + 5 \) at \( x = -1 \).
- Find tangent parallel to x-axis to \( y = x^3 - 6x^2 + 9x + 1 \).
- Find slope of tangent to \( y = \cos x \) at \( x = \pi/3 \).
- Find angle between tangents to \( y = x^2 \) at \( x = 1 \) and \( x = 2 \).
- Find maxima/minima of \( f(x) = \sin x + \sin 2x \).
- Find \( x \) for which \( f(x) = x^3 - 3x + 1 \) is stationary.
- Find derivative of \( y = \ln(x^2 + 1) \).
- Find slope of \( y = e^x \cos x \) at \( x = 0 \).
- Find \( \frac{d^2y}{dx^2} \) if \( y = e^{3x} \).
- Find rate of change of \( y = x^2 \sin x \) at \( x = \pi/3 \).
- Find the tangent to \( y = 1/x \) parallel to line \( y = -x \).
- Find point where tangent to \( y = \sqrt{x} \) has slope \( 1/4 \).
- If \( y = \frac{1}{x^2+1} \), find slope at \( x = 1 \).
- If \( y = e^x + e^{-x} \), find points where \( \frac{dy}{dx} = 0 \).
- Find the rate of change of surface area of a sphere with respect to radius.
- Find rate of change of \( y = x^3 + 2x \) at \( x = 1 \).
- Find \( x \) where \( f(x) = 4x^2 - 8x + 5 \) is minimum.
- Find maxima of \( f(x) = x\sin x \) on \( [0, \pi] \).
- Find the point where \( y = x^2 - 2x + 3 \) has tangent parallel to x-axis.
- Find the tangent to \( y = e^x \) perpendicular to \( y = x \).
- If \( y = \frac{1}{x} \), find rate of change at \( x = 2 \) when \( \frac{dx}{dt} = 3 \).
- Find slope of tangent to \( y = \tan^{-1}(x) \) at \( x = 1 \).
- Find angle between curves \( y = x^2 \) and \( y = x^3 \) at \( x = 1 \).
- Find local minima of \( f(x) = x^3 - 3x + 1 \).
- Find the value of \( k \) so that the tangent to \( y = kx^2 + 2x + 3 \) at \( x = 1 \) is parallel to line \( y = 4x + 5 \).
- Find the point on \( y = \frac{1}{x} \) where tangent is parallel to line \( y = -x \).
- For \( f(x) = 2x^3 - 9x^2 + 12x + 5 \), find all points of local maxima and minima.
- Show that the function \( f(x) = x^3 - 3x^2 + 6x - 2 \) is always increasing.
- Find the value of \( x \) for which the tangent to \( y = \ln x \) passes through the origin.
- The area of a square increases at rate of \( 32\,cm^2/s \). Find rate of change of its side when side = 8 cm.
- If \( y = \sin^{-1}(x^2) \), find \( \frac{dy}{dx} \) at \( x = \frac{1}{2} \).
- Find the maximum and minimum values of \( f(x) = 3\sin x + 4\cos x \).
- Find the points where tangent to \( y = x^3 - 6x^2 + 9x + 1 \) is parallel to x-axis.
- Find the equation of normal to \( y = e^{2x} \) at \( x = 0 \).
🧮Answers
Click to View Answers
- Slope = 5
- Tangent: \( y = 12x - 16 \)
- Normal: \( y = -\frac{1}{6}x + \frac{9}{2} \)
- \( x = 1 \)
- \( \frac{dy}{dx} = \cos x = \frac{1}{\sqrt{2}} \)
- Tangent: \( y = x + 1 \)
- Slope = \( \frac{1}{e} \)
- Slope = 2
- \( x = -\frac{1}{2} \)
- Normal: \( y = -\frac{1}{4}x + 4 \)
- Increasing in (0,3), Decreasing in (3,∞)
- Maxima at \( x = \pi/4 \)
- Min at (2, 2)
- Max at (1, 4)
- Min value = 2 at \( x = 1 \)
- Turning points: \( x = 2, 3 \)
- Inflection at \( x = 2 \)
- Max = 5
- Fastest at \( x = \pi/4 \)
- \( \tan 45 = 1 \Rightarrow x = e^1 = e \)
- \( \frac{dA}{dt} = 2\pi r \frac{dr}{dt} = 42\pi \,cm^2/s \)
- \( \frac{dV}{dt} = 3a^2 \frac{da}{dt} = 150\,cm^3/s \)
- \( \frac{dy}{dx} = x^x(1+\ln x) \)
- Approx. \( \Delta y = \frac{1}{10\sqrt{25}} = 0.01 \)
- Approx. \( \sqrt{100.4} \approx 10.02 \)
- Approx. \( (2.01)^5 ≈ 32.5 \)
- Min of \( x + 1/x \) = 2 at \( x = 1 \)
- Max of \( x(10-x)^2 \) = 296/27
- Inflection \( x = 2 \)
- And so on…
- (101) \( k = 1 \)
- (102) \( x = 1 \)
- (103) Max at \( x = 2 \), Min at \( x = 1 \)
- (104) Always increasing since \( f'(x) = 3x^2 - 6x + 6 > 0 \)
- (105) \( x = e \)
- (106) \( \frac{ds}{dt} = \frac{1}{2\sqrt{A}} \frac{dA}{dt} = 2\,cm/s \)
- (107) \( \frac{dy}{dx} = \frac{2x}{\sqrt{1 - x^4}} = \frac{1}{\sqrt{3}} \)
- (108) Max = 5, Min = -5
- (109) \( x = 1 \text{ and } x = 3 \)
- (110) Normal: \( y = -\frac{1}{2}x + 1 \)
