Class 12 Statistics Notes · GSEB

Definition of Derivative

Definition of Derivative — understand differentiation as rate of change, the limit definition of the derivative, and its tangent-slope meaning. GSEB Class 12 Commerce Statistics notes with interactive graphs.

Last updated: 24 Sep 2026

Notes

What Is Differentiation?

Differentiation

A technique for analysing how a function f(x) changes — how rapidly it changes and by how much — when x changes. The process of obtaining the derivative of a function is called differentiation.

In business you already meet functions like production cost, revenue and profit. Differentiation answers questions like how fast is cost rising right now? or how much extra revenue does the 101st unit bring? — questions averages cannot answer.

Speedometer analogy

Your car's speedometer does not say “average speed since start” — it shows speed at this instant. Differentiation is the speedometer of a function: it reads the instantaneous rate of change at any point.

Definition of Derivative

Derivative at x = a

Let f : A → R and a ∈ A, where A is an open interval of R. If lim(h→0) [f(a + h) − f(a)] / h exists, this limit is called the derivative of f at x = a, denoted f′(a).

★ Definition of Derivative

f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

Derivative as a Function of x

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}, \qquad \frac{dy}{dx} \text{ if } y = f(x)

f′(x)

prime notation

dy/dx

Leibniz notation

ẏ

dot notation

df/dx

operator form

Two words, two meanings

Derivative = the result (a number or function). Differentiation = the process of finding it. Exam questions use both — read carefully.

From Average Change to Instant Change

Take y = 2x² + 3 at x = 2 (y = 11). Nudge x closer and closer to 2 and watch δy/δx — the incrementary ratio — settle on one number.

Incrementary Ratio Lab — y = 2x² + 3 near x = 2

x	δx	y = f(x)	δy	δy / δx
2.1	0.1	11.82	0.82	8.2
2.01	0.01	11.0802	0.0802	8.02
2.001	0.001	11.008	0.008	8.002
2.0001	0.0001	11.0008	0.0008	8.0002

δx = 0.0001 → δy / δx =

8.0002

As δx shrinks toward 0, the ratio walks toward 8 — that is dy/dx at x = 2.

The whole idea in one line

δy itself slides to 0, but the ratio δy/δx settles on a finite number. That limit is the derivative.

Geometry: Slope of the Tangent

Geometrically, f′(x) is the slope of the tangent drawn to the curve at that point. Drag the point below and read the slope in real time.

Tangent Line Explorer — y = x²

Drag the point P along the curve — x = 1.50

Point P

(1.50, 2.25)

Slope = 2x

3.00

Verdict

↗ Rising — positive slope

Watch the tangent rotate

Slide past x = 0: the tangent flips from rising to flat to falling. At x = 0 it lies horizontal — the derivative of x² is 0 at the bottom of the bowl. That is the whole idea of a derivative in one drag.

Geometric Meaning

\frac{dy}{dx} = \text{slope of tangent at the point}

Chai stall connection

A chai seller knows the day's average sale (total ÷ hours). Differentiation tells her the sale rate right now — whether the 8 pm rush is adding ₹400/hour or ₹150/hour — so she can decide when to send the helper for more milk. That is rate of change with money on the line.

Derivative by First Principle — Worked Steps

Exam pattern: substitute into the limit, simplify, cancel h, then let h → 0.

Solved Example

Problem

Obtain the derivative of f(x) = x using the definition.

Solution

f′(x) = 1

Solved Example

Problem

Obtain the derivative of f(x) = x³ using the definition.

Solution

f′(x) = 3x²

Solved Example

Problem

Obtain the derivative of f(x) = k (k a constant) using the definition.

Solution

f′(x) = 0 — the derivative of a constant is always zero

Quick Notation Reference

Notation you will see in the GSEB textbook
Symbol	Reads as	Means
f′(a)	f-prime at a	derivative of f at the point x = a
dy/dx	dee-y by dee-x	rate of change of y as x changes
δy/δx	delta-y by delta-x	average (incrementary) ratio over a finite nudge
h → 0	h tends to zero	the nudge shrinks until only the instant remains

Key Takeaways

Differentiation finds how fast a function changes; the result is called the derivative.
★ f′(a) = lim(h→0) [f(a+h) − f(a)] / h — the first-principle definition you must write in exams.
Geometrically the derivative is the slope of the tangent at that point — drag a tangent and you have the intuition.
δy → 0 as δx → 0, yet δy/δx settles on a finite number — that limit is dy/dx.
Notations f′(x), dy/dx, ẏ and df/dx all mean the same thing — like knowing ₹ and rupees both mean money.
Real use: marginal cost, marginal revenue and profit optimisation in the next topics all start from this definition.

Definition of Derivative

Notes

What Is Differentiation?

Definition of Derivative

From Average Change to Instant Change

Geometry: Slope of the Tangent

Derivative by First Principle — Worked Steps

Quick Notation Reference

Key Takeaways

Related Questions

Study Simplify Special