Application of Derivatives

part 1

Rate of Change and Increasing/Decreasing

• Rate of change of quantities
• Increasing and decreasing functions
• Finding intervals of increase and decrease

part 2

Tangents, Normals and Approximations

• Equation of tangent and normal to a curve
• Angle between two curves
• Approximations using differentials

part 3

Maxima and Minima

• Local maxima and local minima
• First derivative test
• Second derivative test
• Absolute maxima and minima
• Applications of maxima and minima

Increasing on (a,b)

$f'(x) > 0 \;\forall x \in (a,b)$

★ board favourite

Decreasing on (a,b)

$f'(x) < 0 \;\forall x \in (a,b)$

★ board favourite

First derivative test (maxima/minima)

$f'(c)=0$; +→− gives max; −→+ gives min

★ board favourite

Second derivative test

$f'(c)=0$ & $f''(c)<0$ → max; $f''(c)>0$ → min

★ board favourite

Equation of tangent at (x₁,y₁)

$y - y_1 = f'(x_1)(x - x_1)$

★ board favourite

Rate of change

$\frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}$

★ frequently asked

Equation of normal at (x₁,y₁)

$y - y_1 = -\frac{1}{f'(x_1)}(x - x_1)$

★ frequently asked

Small approximations

$\Delta y \approx dy = f'(x)\cdot\Delta x$

★ frequently asked

Show that a right circular cylinder of given volume that is open at the top has minimum total surface area when its height is equal to the radius of its base.

Find the intervals in which the function f(x) = 2x³ − 3x² − 36x + 7 is (a) strictly increasing and (b) strictly decreasing.

Show that the rectangle of maximum area that can be inscribed in a circle of radius r is a square. Find the dimensions of this square.

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall at 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

Find the equation of the tangent and normal to the curve y = x³ − 3x + 2 at the point (2, 4).

Show that the height of a closed right circular cylinder of given surface area and maximum volume is equal to the diameter of its base.

The radius of a circle is increasing at 0.7 cm/s. At what rate is its area increasing when r = 5 cm?

Find the maximum and minimum values of the function f(x) = sin x + cos x on the interval [0, π/2].

Using differentials, find the approximate value of √49.5.

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window that admit the maximum light through the whole opening.

Find the slope of the tangent to the curve y = x³ − x at x = 2.

How many marks is Application of Derivatives worth in the Class 12 board exam?

Around 8 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Maxima/minima word problems — box, cone, rectangle types.

Is Application of Derivatives easy or hard?

It's rated hard — one of the tougher chapters in Class 12 Maths. Most students need about 5 hours to cover it well.

What are the important topics in Application of Derivatives?

The chapter covers 10 NCERT topics in 3 areas: Rate of Change and Increasing/Decreasing; Tangents, Normals and Approximations; Maxima and Minima.

What questions come from Application of Derivatives in board exams?

Between 2018–2024, CBSE board papers asked questions from this chapter on Maxima and Minima, Increasing and Decreasing, Rate of Change, Tangents and Normals — 1- to 5-mark questions.