MathematicsUnit 219 min read

Derivatives: Definition, Rules, Applications & Graphs

Unit 21 of Mathematics explains how to find derivatives (instantaneous rates of change) using first principles, differentiation rules, and applications in real-world problems like optimization and curve sketching.

Padhai · Based on the official syllabus and past papers
Updated 26 Sept 2026

TAKEAWAYS:

Derivatives measure how fast a function changes at any point (slope of the tangent line).
Basic rules (power, product, quotient, chain) simplify differentiation without using first principles.
Graphs help visualize where functions increase/decrease, have maxima/minima, or change concavity.
Applications include finding tangents, rates of change, and optimizing quantities (e.g., profit, area).
Common mistakes: Forgetting the chain rule for composite functions or misapplying power rules to negative exponents.

1. Introduction to Derivatives

A derivative tells us how fast a quantity changes at a specific point. For example:

If $s (t)$ is the position of a car at time $t$ , then $s^{'} (t)$ is its instantaneous speed.
If $P (x)$ is the profit of a company based on sales $x$ , then $P^{'} (x)$ tells us how profit changes with each additional sale.

Illustration of the limit definition: Δy/Δx as h approaches 0.

Geometric interpretation of the derivative as the slope of the tangent line to y = x² at x = 1 and x = 2.

1.1 Definition: Derivative as a Limit

The derivative of a function $f (x)$ at a point $x$ is defined as: $f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}$ This is called the limit definition or first principles.

1.2 Geometric Meaning

The derivative $f^{'} (x)$ is the slope of the tangent line to the curve $y = f (x)$ at $x$ .

1.3 Example: Find $f^{'} (x)$ for $f (x) = x^{2}$ using first principles

Step 1: Write the limit definition: $f^{'} (x) = h \to 0 lim \frac{( x + h ) ^{2} - x ^{2}}{h}$ Step 2: Expand and simplify: $= h \to 0 lim \frac{x ^{2} + 2 x h + h ^{2} - x ^{2}}{h} = h \to 0 lim \frac{2 x h + h ^{2}}{h} = h \to 0 lim (2 x + h)$ Step 3: Take the limit: $f^{'} (x) = 2 x$

2. Basic Differentiation Rules

Instead of using first principles every time, we use shortcut rules:

Graph of f(x) = x³ − 3x² with its derivative f'(x) to show critical points.

Rule	Formula	Example
Power Rule	$\frac{d}{d x} x^{n} = n x^{n - 1}$	$\frac{d}{d x} x^{3} = 3 x^{2}$
Constant Rule	$\frac{d}{d x} c = 0$	$\frac{d}{d x} 5 = 0$
Constant Multiple	$\frac{d}{d x} [c f (x)] = c f^{'} (x)$	$\frac{d}{d x} 3 x^{2} = 6 x$
Sum/Difference	$\frac{d}{d x} [f (x) \pm g (x)] = f^{'} (x) \pm g^{'} (x)$	$\frac{d}{d x} (x^{2} + x) = 2 x + 1$
Product Rule	$\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)$	$\frac{d}{d x} (x^{2} \cdot sin x) = 2 x sin x + x^{2} cos x$
Quotient Rule	$\frac{d}{d x} (\frac{f ( x )}{g ( x )}) = \frac{f ^{'} ( x ) g ( x ) - f ( x ) g ^{'} ( x )}{[ g ( x ) ] ^{2}}$	$\frac{d}{d x} (\frac{x}{x + 1}) = \frac{1}{( x + 1 ) ^{2}}$
Chain Rule	$\frac{d}{d x} f (g (x)) = f^{'} (g (x)) \cdot g^{'} (x)$	$\frac{d}{d x} sin (2 x) = 2 cos (2 x)$

2.1 Example: Differentiate $f (x) = 3 x^{4} - 2 x^{3} + 5 x - 7$

Using the power rule and sum rule: $f^{'} (x) = 3 \cdot 4 x^{3} - 2 \cdot 3 x^{2} + 5 \cdot 1 - 0 = 12 x^{3} - 6 x^{2} + 5$

2.2 Example: Differentiate $f (x) = (2 x^{2} + 3) (x^{3} - 1)$ (Product Rule)

Let $u = 2 x^{2} + 3$ , $v = x^{3} - 1$ . Then: $u^{'} = 4 x, v^{'} = 3 x^{2}$ Using the product rule: $f^{'} (x) = u^{'} v + u v^{'} = (4 x) (x^{3} - 1) + (2 x^{2} + 3) (3 x^{2})$ Simplify: $= 4 x^{4} - 4 x + 6 x^{4} + 9 x^{2} = 10 x^{4} + 9 x^{2} - 4 x$

2.3 Example: Differentiate $f (x) = \frac{x ^{2}}{x + 1}$ (Quotient Rule)

Let $u = x^{2}$ , $v = x + 1$ . Then: $u^{'} = 2 x, v^{'} = 1$ Using the quotient rule: $f^{'} (x) = \frac{u ^{'} v - u v ^{'}}{v ^{2}} = \frac{2 x ( x + 1 ) - x ^{2} ( 1 )}{( x + 1 ) ^{2}} = \frac{2 x ^{2} + 2 x - x ^{2}}{( x + 1 ) ^{2}} = \frac{x ^{2} + 2 x}{( x + 1 ) ^{2}}$

2.4 Example: Differentiate $f (x) = sin (3 x)$ (Chain Rule)

Let $u = 3 x$ , $f (u) = sin u$ . Then: $f^{'} (u) = cos u, u^{'} = 3$ Using the chain rule: $f^{'} (x) = cos (3 x) \cdot 3 = 3 cos (3 x)$

3. Derivatives of Common Functions

Here’s a quick reference table:

Function	Derivative
$sin x$	$cos x$
$cos x$	$- sin x$
$tan x$	$sec^{2} x$
$e^{x}$	$e^{x}$
$ln x$	$\frac{1}{x}$
$a^{x}$	$a^{x} ln a$
$lo g_{a} x$	$\frac{1}{x l n a}$

4. Applications of Derivatives

Derivatives help solve real-world problems:

Profit function P(x) = −2x² + 100x − 1000 with its maximum at x = 25.

4.1 Finding Tangents

Find the equation of the tangent line to $y = x^{2}$ at $x = 2$ . Step 1: Find $f^{'} (x) = 2 x$ . At $x = 2$ , slope $m = 4$ . Step 2: Point is $(2, 4)$ . Use point-slope form: $y - 4 = 4 (x - 2) ⟹ y = 4 x - 4$

4.2 Rates of Change

If $A (t) = t^{2} + 2 t$ is the area of a square at time $t$ , find how fast the area changes at $t = 3$ . $A^{'} (t) = 2 t + 2 ⟹ A^{'} (3) = 8$ So, the area increases at 8 square units per second at $t = 3$ .

4.3 Optimization (Maxima/Minima)

Find the maximum profit if $P (x) = - 2 x^{2} + 100 x - 1000$ . Step 1: Find $P^{'} (x) = - 4 x + 100$ . Step 2: Set $P^{'} (x) = 0$ : $- 4 x + 100 = 0 ⟹ x = 25$ Step 3: Check concavity with $P^{''} (x) = - 4$ (negative, so maximum). Maximum profit at $x = 25$ : $P (25) = - 2 (25)^{2} + 100 (25) - 1000 = 1125$

4.4 Increasing/Decreasing Functions

A function is:

Increasing where $f^{'} (x) > 0$ .
Decreasing where $f^{'} (x) < 0$ .

For $f (x) = x^{3} - 3 x^{2}$ : $f^{'} (x) = 3 x^{2} - 6 x = 3 x (x - 2)$

Increasing when $x < 0$ or $x > 2$ .
Decreasing when $0 < x < 2$ .

5. Higher-Order Derivatives

The second derivative $f^{''} (x)$ tells us about concavity:

If $f^{''} (x) > 0$ , the curve is concave up (like $\cup$ ).
If $f^{''} (x) < 0$ , the curve is concave down (like $\cap$ ).

Concavity changes at x = 1 where f''(x) = 0 (inflection point).

For $f (x) = x^{3} - 3 x^{2}$ : $f^{'} (x) = 3 x^{2} - 6 x, f^{''} (x) = 6 x - 6$

Concave up when $x > 1$ .
Concave down when $x < 1$ .

6. Exam Tips

Memorize basic rules: Power, product, quotient, and chain rules are frequently tested.
Practice first principles: Even though rules are faster, NEB sometimes asks for the limit definition.
Check for critical points: Always set $f^{'} (x) = 0$ to find maxima/minima.
Graph interpretation: Sketch the graph and label increasing/decreasing intervals and concavity.
Units matter: In applied problems, always include units (e.g., "meters per second").
Common mistakes:
- Forgetting the chain rule for composite functions (e.g., $sin (2 x)$ → $2 cos (2 x)$ ).
- Misapplying the quotient rule (denominator must be squared).
- Ignoring domain restrictions (e.g., $\frac{1}{x}$ is undefined at $x = 0$ ).

NEB Board-Style Questions

Short Answer (5 marks each)

Find the derivative of $f (x) = \frac{x ^{2} + 1}{x - 1}$ using the quotient rule.
Differentiate $f (x) = e^{2 x} sin x$ using the product rule and chain rule.
Find the equation of the tangent line to $y = x$ at $x = 4$ .
Determine where $f (x) = x^{3} - 6 x^{2} + 9 x$ is increasing and decreasing.
If $s (t) = t^{2} - 4 t + 3$ is the position of a particle, find its velocity at $t = 2$ .

Long Answer (10 marks each)

A company’s profit function is $P (x) = - x^{2} + 100 x - 1000$ , where $x$ is the number of units sold.
- Find the number of units that maximizes profit.
- Calculate the maximum profit.
- Sketch the graph of $P (x)$ and label the critical point.
Differentiate $f (x) = \frac{( x ^{2} + 1 ) ( x - 2 )}{x}$ from first principles and simplify.
For $f (x) = x^{4} - 4 x^{3} + 6 x^{2}$ :
- Find $f^{'} (x)$ and $f^{''} (x)$ .
- Determine where the function is concave up/down.
- Find all critical points and classify them as maxima/minima.

tangent line to a curve A tangent line touches the curve at exactly one point and has the same slope as the curve there. (Image: Jacj at English Wikipedia / Later versions were uploaded by , Public domain, via Wikimedia Commons)

Based on the NEB +2 Science syllabus for Mathematics (Maths), unit 21.

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