ClassApna

Class 12 Maths Notes

Application of Derivatives Class 12 Notes

Exam-ready notes on applying derivatives: instantaneous rate of change, increasing and decreasing functions, tangent and normal lines, and finding maxima and minima with the first and second derivative tests — with solved examples for boards and JEE.

Class12SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

The derivative measures how fast a quantity changes, and that information decides where a curve rises, falls, has tangents and attains its highest and lowest points.

Rate of Change and Tangents

Rate of change

For y=f(x)y = f(x), the derivative dy/dxdy/dx is the instantaneous rate of change of y with respect to x — speed, velocity, marginal cost and population growth are all examples.

Tangent: y−y1=m(x−x1),m=dydx∣(x1,y1),Normal slope=−1m\text{Tangent: } y - y_1 = m(x - x_1),\quad m = \left.\frac{dy}{dx}\right|_{(x_1,y_1)},\qquad \text{Normal slope} = -\frac{1}{m}
Tangent and normal equations

Increasing and Decreasing Functions

  • f is increasing on an interval where
  • f′(x)>0f'(x) > 0
  • and decreasing where
  • f′(x)<0f'(x) < 0
  • .
  • If f'(x) = 0 throughout an interval, f is constant there.
  • Stationary points satisfy f'(x) = 0 and separate increasing from decreasing regions.

Sign of the derivative

A single point where f'(x) = 0 does not make the function non-monotonic — check the sign of the derivative on each side before concluding anything about maxima or minima.

Maxima and Minima

Critical point

An extremum of f can occur only at a critical point where f′(x)=0f'(x) = 0 or where f' is undefined.

f′′(a)<0⇒local max,f′′(a)>0⇒local min,f′′(a)=0⇒test inconclusivef''(a) < 0 \Rightarrow \text{local max},\qquad f''(a) > 0 \Rightarrow \text{local min},\qquad f''(a) = 0 \Rightarrow \text{test inconclusive}
Second derivative test

Optimisation and Approximations

  • Word problems: identify the quantity to optimise, write it as a function of one variable using the constraints, find critical points, and test them.
  • Approximation:
  • f(x+Δx)≈f(x)+f′(x)Δxf(x + \Delta x) \approx f(x) + f'(x)\Delta x
  • .
  • The largest or smallest value on a closed interval is found by comparing critical points with the endpoints.

Solved Examples

Example: Find the intervals on which f(x)=x3−3x+1f(x) = x^3 - 3x + 1 is increasing.

Solution: f′(x)=3x2−3=3(x−1)(x+1)f'(x) = 3x^2 - 3 = 3(x-1)(x+1). This is positive for x<−1x < -1 and x>1x > 1, so f is increasing on (−∞,−1)∪(1,∞)(-\infty,-1) \cup (1,\infty) and decreasing on (−1,1)(-1,1).

Example: Show that every edge of the largest rectangle of fixed perimeter 20 cm is 5 cm.

Solution: Let sides be x and 10 − x. Area A=x(10−x)=10x−x2A = x(10-x) = 10x - x^2. Then A′=10−2x=0⇒x=5A' = 10 - 2x = 0 \Rightarrow x = 5 and A′′=−2<0A'' = -2 < 0, so x = 5 is a maximum — a square of side 5 cm.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Rate of change

dy/dx=lim⁡Δx→0ΔyΔxdy/dx = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}

Tangent line

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Normal line

y−y1=−1m(x−x1)y - y_1 = -\frac{1}{m}(x - x_1)

Second derivative test

f′′(a)<0⇒max, f′′(a)>0⇒minf''(a) < 0 \Rightarrow \text{max},\ f''(a) > 0 \Rightarrow \text{min}

Linear approximation

f(x+Δx)≈f(x)+f′(x)Δxf(x+\Delta x) \approx f(x) + f'(x)\Delta x

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • Increasing means f'(x) > 0 on an interval, not at a single point.
  • Set f'(x) = 0 to get candidates, then use the second derivative test.
  • On a closed interval compare critical values with the endpoint values.
  • For optimisation word problems, first write the quantity as one function of one variable.
  • The normal is perpendicular to the tangent, so its slope is −1/m.

FAQ

Common questions

What is the difference between the first and second derivative tests?

The first derivative test looks at sign changes of f′ around a critical point; the second derivative test evaluates f″ at it — negative gives a max, positive gives a min, and zero gives no conclusion.

How do you find maxima and minima of a function?

Find critical points where f′(x) = 0, test each with the second derivative (or sign changes), and on a closed interval also check the endpoints.

What does the derivative represent geometrically?

The derivative at a point is the slope of the tangent line to the curve there, which is also the instantaneous rate of change of the function.

When is a function increasing or decreasing?

A function is increasing where its derivative is positive and decreasing where its derivative is negative, over a full interval rather than at isolated points.

Continue learning

Related notes

Continuity and Differentiability NotesIntegrals Class 12 NotesClass 12 Maths NotesMaths Tutor in Mohali

Mastering this chapter with live help

Notes help, but doubts clear fastest in a live class. Narayan Gurukul Academy (ClassApna) runs small-batch CBSE, JEE and NEET coaching from our Mohali centre and online — with daily doubt support and mock tests.

Book a Free DemoSee plans & pricing

One-on-one guidance available · Live online classes across India