Differentiation — Standard Derivatives and the Chain Rule

Q: Do I need to memorize standard derivatives for 0606?

Yes, all standard derivatives covered in this guide are not provided in the exam formula booklet, so you must memorize them for both Papers 1 and 2.

Q: Can I use the chain rule for more than two composite functions?

Yes, the chain rule extends to any number of nested composite functions: work from the outermost function inward, multiplying by the derivative of each inner layer sequentially.

Q: Is differentiation from first principles tested in 0606?

No, differentiation from first principles is explicitly out of scope for CIE IGCSE Additional Mathematics 0606, so you will not be assessed on this technique.

Additional Mathematics· 14.1, 14.2, 14.3· 25 min read

1. Standard Derivatives & Notation★★☆☆☆⏱ 6 min

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📘 Definition

First Derivative

$f^{'} (x)$ , $\frac{d y}{d x}$

The function that gives the gradient of the original function $f (x)$ at any valid point $x$ .

All standard derivatives below must be memorized, as they are not provided in the 0606 exam formula booklet. All trigonometric derivatives apply only when angles are measured in radians.

Function $f (x)$	Derivative $f^{'} (x)$
$x^{n}$ , $n$ rational	$n x^{n - 1}$
$sin x$	$cos x$
$cos x$	$- sin x$
$tan x$	$sec^{2} x$
$e^{x}$	$e^{x}$
$ln x$	$\frac{1}{x}$

📐 Worked Example

Find the first derivative of $y = 4 x^{3} + 2 ln x - 3 cos x$

1
Differentiate each term separately using the standard derivatives table.
2
$\frac{d}{dx}(4x^3) = 4 \times 3x^{2} = 12x^2$
3
$\frac{d}{dx}(2\text{ln }x) = 2 \times \frac{1}{x} = \frac{2}{x}$
4
$\frac{d}{dx}(-3\text{cos }x) = -3 \times (-\text{sin }x) = 3\text{sin }x$
5
Combine all terms to get the final derivative.
6
$\frac{dy}{dx} = 12x^2 + \frac{2}{x} + 3\text{sin }x$

Exam tip:

Always check you have included the negative sign for the derivative of $\frac{d}{dx} (cos x) = - sin x$ — this is one of the most common mark-losing mistakes in Paper 1.

2. Introduction to the Chain Rule★★★☆☆⏱ 7 min

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📘 Definition

Chain Rule

$\frac{d y}{d x} = \frac{d y}{d u} \times \frac{d u}{d x}$ where $u = g (x)$ is the inner function of $y = f (g (x))$

Rule used to differentiate composite functions, by multiplying the derivative of the outer function (keeping the inner function unchanged) by the derivative of the inner function.

📐 Worked Example

Find $\frac{d y}{d x}$ for $y = (2 x + 5)^{7}$ using the chain rule.

1
Identify inner and outer functions: let $u = 2 x + 5$ , so $y = u^{7}$ .
2
$\frac{dy}{du} = 7u^6$
3
$\frac{du}{dx} = 2$
4
Apply the chain rule formula, then substitute back $u = 2 x + 5$ .
5
$\frac{dy}{dx} = 7u^6 \times 2 = 14(2x +5)^6$

✓ Quick check

What is the derivative of $(3 x - 1)^{4}$ ?
- $4 (3 x - 1)^{3}$
- $12 (3 x - 1)^{3}$
- $12 (3 x - 1)^{4}$
- $3 (3 x - 1)^{3}$
Reveal answer
1 —
Correct: Derivative of outer function is $4 (3 x - 1)^{3}$ , derivative of inner function is 3, multiplied gives $12 (3 x - 1)^{3}$ .

3. Chain Rule for Composite Standard Functions★★★★☆⏱ 8 min

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Combine the chain rule with all standard derivatives (trigonometric, exponential, logarithmic) to differentiate any composite function allowed in the 0606 syllabus.

📐 Worked Example

Differentiate $y = e^{sin x}$ with respect to $x$ .

1
Let inner function $u = sin x$ , so $y = e^{u}$ .
2
$\frac{dy}{du} = e^u = e^{\text{sin }x}, \frac{du}{dx} = \text{cos }x$
3
$\frac{dy}{dx} = e^{\text{sin }x} \times \text{cos }x = \text{cos }x e^{\text{sin }x}$

📐 Worked Example

Find $f^{'} (x)$ if $f (x) = ln (tan x)$ where $0 < x < \frac{π}{2}$ .

1
Let inner function $u = tan x$ , so $f (u) = ln u$ .
2
$f'(u) = \frac{1}{u} = \frac{1}{\text{tan }x}, u' = \text{sec}^2 x$
3
$f'(x) = \frac{1}{\text{tan }x} \times \text{sec}^2 x = \frac{1}{\text{sin }x \text{ cos }x}$

Exam tip:

You do not need to simplify trigonometric derivatives further unless explicitly asked to in the question, though simplifying may help with follow-up problem parts.

4. Second Derivatives of Composite Functions★★★★☆⏱ 4 min

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📘 Definition

Second Derivative

$f^{''} (x)$ , $\frac{d ^{2} y}{d x ^{2}}$

The derivative of the first derivative of a function, used to find the nature of stationary points and rates of change of gradients.

📐 Worked Example

Find $\frac{d ^{2} y}{d x ^{2}}$ for $y = cos (3 x)$ .

1
First calculate the first derivative using the chain rule.
2
$\frac{dy}{dx} = -\text{sin}(3x) \times 3 = -3\text{sin}(3x)$
3
Differentiate the first derivative again to get the second derivative, applying the chain rule a second time.
4
$\frac{d^2y}{dx^2} = -3 \times \text{cos}(3x) \times 3 = -9\text{cos}(3x)$

5. Common Pitfalls

Wrong move:

Forgetting the negative sign when differentiating $cos x$

Why:

The derivative of $cos x$ is $- sin x$ , not $sin x$ , so missing the sign costs 1 mark per occurrence.

Correct move:

Memorize trigonometric derivative pairs explicitly: $sin x \to + cos x$ , $cos x \to - sin x$ , $tan x \to + sec^{2} x$ .

Wrong move:

Forgetting to multiply by the derivative of the inner function when applying the chain rule

Why:

The chain rule requires two factors (outer derivative × inner derivative), omitting the inner derivative gives an incorrect result.

Correct move:

Use the 'outer first, inner next' mnemonic to ensure you include both factors for every composite function differentiation.

Wrong move:

Using degrees instead of radians for trigonometric differentiation

Why:

Standard trigonometric derivatives only apply when angles are measured in radians, the default for all 0606 calculus questions.

Correct move:

Always convert any angle given in degrees to radians before differentiating trigonometric functions.

Wrong move:

Differentiating $ln x$ as $ln (\frac{1}{x})$ or $x$ instead of $\frac{1}{x}$

Why:

The derivative of $ln x$ is a reciprocal function, not another logarithmic or linear function.

Correct move:

Add the pair $ln x \to \frac{1}{x}$ to your memorized standard derivatives and test yourself regularly.

Wrong move:

Expanding high-power composite functions (e.g. $(2 x + 7)^{9}$ ) before differentiating

Why:

Expanding these functions is time-consuming and prone to arithmetic errors, wasting valuable exam time.

Correct move:

Always use the chain rule for composite power functions, no matter how high the exponent.

6. Quick Reference Cheatsheet

Rule/Function	Derivative Result
$x^{n}$ (n rational)	$n x^{n - 1}$
$sin x$	$cos x$
$cos x$	$- sin x$
$tan x$	$sec^{2} x$
$e^{x}$	$e^{x}$
$ln x$	$\frac{1}{x}$
Chain Rule ( $y = f (g (x))$ )	$f^{'} (g (x)) \times g^{'} (x)$
Second Derivative	Derivative of first derivative

7. Frequently Asked

Do I need to memorize standard derivatives for 0606?

Yes, all standard derivatives covered in this guide are not provided in the exam formula booklet, so you must memorize them for both Papers 1 and 2.

Can I use the chain rule for more than two composite functions?

Yes, the chain rule extends to any number of nested composite functions: work from the outermost function inward, multiplying by the derivative of each inner layer sequentially.

Is differentiation from first principles tested in 0606?

No, differentiation from first principles is explicitly out of scope for CIE IGCSE Additional Mathematics 0606, so you will not be assessed on this technique.

Going deeper

Syllabus DocumentCIE IGCSE Add Maths 0606 2025-2027 SyllabusOfficial syllabus reference for calculus section

What's Next

Now that you have mastered standard derivatives and the chain rule, you are ready to learn the product and quotient rules for differentiating products and fractions of functions, followed by applications of differentiation including finding stationary points, rates of change, and optimization problems for CIE IGCSE Additional Mathematics 0606. These skills are heavily tested across both Papers 1 and 2, so practice with a wide range of composite function questions to build speed and accuracy before your exam.