Differentiation Rules — SL 5.3 Differentiation rules (power, sin, cos, e^x, ln x; product/quotient/chain) (as included) - IB Mathematics Analysis and Approaches

SECTION 1

Historical Context & Motivation

For centuries, mathematicians and scientists struggled with a fundamental question: how do you measure the rate at which something changes at a single instant? If a ball is rolling down a hill, you can measure its average speed over ten seconds, but what about its speed at exactly the three-second mark? This puzzle drove the development of calculus, and at the heart of calculus lies differentiation — the process of finding how fast a function's output changes relative to its input.

Early differentiation relied on going back to first principles every single time, computing limits of difference quotients for each new function. This was tedious and error-prone. Mathematicians gradually discovered shortcuts — differentiation rules — that let you find derivatives quickly and reliably. These rules form the backbone of IB Math AA SL 5.3 and are among the most important tools you'll ever learn in mathematics.

1665–1676 • Newton & Leibniz Develop Calculus

Isaac Newton and Gottfried Wilhelm Leibniz independently developed the fundamental ideas of calculus, including the concept of the derivative. Newton called them "fluxions," while Leibniz introduced the dy/dx notation still used today.

1748 • Euler's Introductio

Leonhard Euler published systematic treatments of exponential and trigonometric functions, establishing the derivatives of eˣ, sin x, and cos x as standard results that every student could apply.

1797 • Lagrange Formalises the Derivative

Joseph-Louis Lagrange introduced the prime notation f′(x) and emphasized that differentiation rules — power, product, quotient, and chain — could be stated as general theorems applicable to broad classes of functions.

1821 • Cauchy's Rigorous Limits

Augustin-Louis Cauchy placed the derivative on a rigorous footing using the formal limit definition, confirming that the differentiation rules discovered by earlier mathematicians were logically airtight.

The question these mathematicians answered was practical: can we find the derivative of any function without going back to the limit definition every time? The answer is yes — and the rules in SL 5.3 are precisely those shortcuts. Once you learn them, you can differentiate polynomials, trigonometric expressions, exponentials, logarithms, and complex combinations of these functions in seconds rather than minutes.

SECTION 2

Core Principles & Definitions

Before diving into individual rules, it helps to understand the big picture. The derivative of a function f(x), written f′(x) or dy/dx, tells you the instantaneous rate of change of f at any point x. Every differentiation rule is simply a proven shortcut that lets you compute f′(x) without returning to the limit definition.

Basic Derivative Rules

The power rule, derivatives of sin x, cos x, eˣ, and ln x form the foundation. These are the building blocks for every other derivative you'll compute.

Linearity of Derivatives

You can differentiate term by term: the derivative of a sum is the sum of the derivatives, and constants can be pulled out front. This means d/dx [af(x) + bg(x)] = af′(x) + bg′(x).

Product & Quotient Rules

When two functions are multiplied or divided, you cannot simply differentiate each piece separately. The product rule and quotient rule handle these situations.

Chain Rule

When a function is composed inside another function — like sin(3x²) — you need the chain rule. It tells you to differentiate the "outer" function and multiply by the derivative of the "inner" function.

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KEY TAKEAWAY

Think of differentiation rules like a mechanic's toolbox. The power rule is your wrench — you'll use it constantly. The product, quotient, and chain rules are your specialty tools for trickier jobs. Just as a mechanic doesn't rebuild the engine from scratch to change a spark plug, you don't go back to the limit definition for every derivative. You pick the right tool and apply it.

SECTION 3

Visual Explanation — The Power Rule in Action

The diagram below shows how the power rule connects a function to its derivative. On the left you see f(x) = x³ (the original curve), and on the right you see f′(x) = 3x² (the derivative). Notice that wherever the original function is climbing steeply, the derivative has a large positive value, and wherever the original function levels off, the derivative crosses zero.

Left: the cubic function f(x) = x³ passes through the origin, decreasing for negative x and increasing for positive x. Right: its derivative f′(x) = 3x² is always non-negative, touching zero at x = 0 where the cubic briefly flattens. The yellow arrow represents the operation of differentiation — applying the power rule to transform the original into its rate-of-change function.

This visual pattern generalises to every function you'll meet. The derivative captures the slope of the tangent line at each point. When the original curve is steep, the derivative is large. When it's flat, the derivative is zero. When the curve is falling, the derivative is negative. Understanding this connection visually will help you check your algebraic work — if your derivative doesn't match the shape you'd expect, you know to look for a mistake.

SECTION 4

Mathematical Framework — The Core Rules

Here are the fundamental differentiation rules you need for IB Math AA SL 5.3. Each rule tells you how to find the derivative of a particular type of function. Memorise these — they are given in the IB formula booklet, but speed comes from knowing them by heart.

Basic Derivative Rules

POWER RULE

d/dx [xⁿ] = nxⁿ⁻¹

If f(x) = xⁿ where n is any real number, bring the exponent down as a coefficient and reduce the exponent by 1. For example, d/dx [x⁵] = 5x⁴.

TRIGONOMETRIC RULES

d/dx [sin x] = cos x d/dx [cos x] = −sin x

The derivative of sine is cosine, and the derivative of cosine is negative sine. The negative sign on the cosine derivative is a common source of errors — watch for it!

EXPONENTIAL RULE

d/dx [eˣ] = eˣ

The natural exponential function eˣ is its own derivative. This remarkable property is what makes e ≈ 2.718 so special in mathematics.

NATURAL LOGARITHM RULE

d/dx [ln x] = 1/x

The derivative of the natural logarithm is the reciprocal function 1/x. This is defined only for x > 0, just as ln x itself is only defined for positive x.

Combination Rules

PRODUCT RULE

d/dx [u · v] = u′v + uv′

If y = u(x) · v(x), then y′ = u′(x) · v(x) + u(x) · v′(x). Differentiate the first, multiply by the second, then add the first times the derivative of the second.

QUOTIENT RULE

d/dx [u/v] = (u′v − uv′) / v²

If y = u(x)/v(x), the derivative is (u′v − uv′)/v². Remember: it's "low d-high minus high d-low, over low squared." The subtraction order matters — it's not commutative.

CHAIN RULE

d/dx [f(g(x))] = f′(g(x)) · g′(x)

For a composite function, differentiate the outer function (evaluated at the inner function) and multiply by the derivative of the inner function. In Leibniz notation: dy/dx = (dy/du) × (du/dx).

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IB Tip

The IB formula booklet provides these rules, so you don't need to derive them from scratch on the exam. However, you must know when to apply each one. The chain rule is often used in combination with the product or quotient rule — practice recognising which rules are needed before you start writing.

SECTION 5

Choosing the Right Rule — A Decision Flowchart

One of the biggest challenges students face is not the rules themselves, but knowing which rule to use. The flowchart below walks you through the decision process. Start at the top and follow the arrows based on the structure of your function.

Decision flowchart for selecting a differentiation rule. Start at the top with your function's structure. Note the dashed box at the bottom left: many IB exam questions require combining rules — for example, using the chain rule inside a product rule.

Function Type	Example	Rule(s) Needed
Simple power	f(x) = 4x⁷	Power rule
Single trig / exp / log	f(x) = 3sin x	Basic rule + constant multiple
Product of two functions	f(x) = x² · eˣ	Product rule
Fraction of two functions	f(x) = sin x / x	Quotient rule
Function inside a function	f(x) = eˢⁱⁿ ˣ	Chain rule
Combined	f(x) = x · sin(3x)	Product rule + chain rule

Common function types and the rules required to differentiate them

SECTION 6

Worked Example — Differentiating a Combined Function

Let's work through a multi-rule problem step by step. Suppose we need to find the derivative of f(x) = x² · sin(3x). This involves both the product rule and the chain rule.

Find f′(x) where f(x) = x² · sin(3x)

1
Step 1 — Identify the Structure

The function is a product of two functions: u(x) = x² and v(x) = sin(3x). We will need the product rule: (uv)′ = u′v + uv′. Notice that v(x) = sin(3x) is itself a composite function (sine of 3x), so we'll need the chain rule when we differentiate v.
2
Step 2 — Find u′(x)

Using the power rule on u(x) = x²:

u′(x) = 2x
3
Step 3 — Find v′(x) Using the Chain Rule

For v(x) = sin(3x), the outer function is sin(·) and the inner function is 3x. The derivative of sin(·) is cos(·), and the derivative of 3x is 3. By the chain rule: v′(x) = cos(3x) × 3.

v′(x) = 3cos(3x)
4
Step 4 — Apply the Product Rule

Now substitute into the product rule formula: f′(x) = u′v + uv′ = (2x)(sin(3x)) + (x²)(3cos(3x)).

f′(x) = 2x sin(3x) + 3x² cos(3x)
5
Step 5 — Verify and Simplify

We can factor out x if desired: f′(x) = x[2sin(3x) + 3x cos(3x)]. This is the final answer. As a quick check, when x = 0, f(0) = 0 and f′(0) = 0, which makes sense because the x² factor forces the function and its derivative to be zero at the origin.

f′(x) = x[2sin(3x) + 3x cos(3x)]

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Common Mistake Alert

Many students forget the chain rule inside the product rule. They correctly set up u′v + uv′ but then write the derivative of sin(3x) as cos(3x) instead of 3cos(3x). Always ask yourself: "Is there an inner function that needs differentiating?"

SECTION 7

When to Use Each Rule — Strengths & Pitfalls

Each differentiation rule has its own strengths and common pitfalls. Understanding these will help you avoid the mistakes that cost marks on the IB exam.

Rule	Strengths	Common Pitfalls
Power Rule	Fast, applies to any real exponent including fractions and negatives (e.g., $x^{1/2}$ or $x^{−1}$ )	Forgetting to rewrite roots and reciprocals as powers first: √x = $x^{1/2}$ , 1/x = $x^{−1}$
Trig Rules	Direct, no rearranging needed; pattern is cyclic (sin → cos → −sin → −cos)	Dropping the negative sign in d/dx [cos x] = −sin x; confusing sin and cos derivatives
eˣ and ln x	eˣ is its own derivative — simplest rule to apply; ln x gives a clean reciprocal	Applying these rules to aˣ or $\log_a$ (x) without using change-of-base or chain rule
Product Rule	Works for any product of two functions; formula is symmetric in structure	Writing u′ × v′ instead of u′v + uv′; forgetting that constants still count (e.g., 5eˣ is 5 × eˣ)
Quotient Rule	Handles any ratio of functions systematically	Getting the subtraction order wrong (it must be u′v − uv′, not uv′ − u′v); forgetting to square the denominator
Chain Rule	Essential for composite functions; can be chained multiple times for deeply nested compositions	Forgetting the inner derivative entirely; this is the #1 error in IB differentiation questions

Comparison of differentiation rules: strengths and common pitfalls

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KEY TAKEAWAY

Think of the chain rule as peeling an onion. Each layer is a function wrapped around another. You differentiate the outermost layer first, then peel inward, multiplying by the derivative of each inner layer as you go. If you have sin(

e^{2x}

), that's three layers: sin(·),

e^{·}

, and 2x. You need to multiply three derivatives together: cos(

e^{2x}

) ×

e^{2x}

× 2.

SECTION 8

Connection to Higher-Level Calculus

The differentiation rules you've learned in SL 5.3 are the foundation for everything that comes next in calculus. Whether you continue to HL Mathematics or encounter calculus in university, these rules are the tools you'll always rely on.

SL 5.3 Concept	Where It Leads (HL / University)
Power rule for xⁿ	Generalised power rule for [f(x)]ⁿ using the chain rule; implicit differentiation where y is a function of x
Derivatives of sin x, cos x	Derivatives of tan x, sec x, csc x, cot x; inverse trig derivatives; Taylor series for trig functions
Derivative of eˣ	Derivative of aˣ = aˣ ln a; differential equations involving exponential growth and decay
Product and quotient rules	Integration by parts (which is the reverse of the product rule); partial fractions for integration
Chain rule	Related rates; u-substitution in integration; multivariable chain rule in partial derivatives

How SL 5.3 differentiation rules connect to advanced topics

If you pursue HL Mathematics, you'll also encounter implicit differentiation, where y is not written as an explicit function of x. Even there, the core rules are the same — you just apply them while treating y as a function of x and multiplying by dy/dx whenever you differentiate a y term (that's the chain rule in disguise!). The message is clear: mastering these rules now sets you up for every calculus topic you'll ever meet.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain in your own words why the derivative of eˣ is special compared to the derivative of other exponential functions like 2ˣ or 10ˣ. What property makes e unique?

Show answer

The function eˣ is the only exponential function that is its own derivative: d/dx [eˣ] = eˣ. For other bases, d/dx [aˣ] = aˣ · ln a, which introduces an extra constant factor. The number e ≈ 2.718 is defined precisely so that this constant factor equals 1 (since ln e = 1). This self-replicating property is what makes e the "natural" base for exponential functions in calculus, and it's why eˣ appears so frequently in growth and decay models.

Question 2: BASIC CALCULATION

Find the derivative of f(x) = 5x⁴ − 3cos x + ln x.

Show answer

Apply each basic rule term by term. For 5x⁴, the power rule gives 5 × 4x³ = 20x³. For −3cos x, the derivative of cos x is −sin x, so we get −3(−sin x) = 3sin x. For ln x, the derivative is 1/x. Combining: f′(x) = 20x³ + 3sin x + 1/x. Each term was differentiated independently using the linearity of the derivative, and no product, quotient, or chain rule was needed because the terms are simply added together.

Question 3: INTERMEDIATE

Differentiate g(x) = e²ˣ · cos x using the product rule and chain rule.

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Set u = e²ˣ and v = cos x. First, find u′ using the chain rule: the outer function is

e^{·}

with derivative

e^{·}

, and the inner function is 2x with derivative 2, so u′ = 2e²ˣ. Next, v′ = −sin x. Apply the product rule: g′(x) = u′v + uv′ = 2e²ˣ cos x + e²ˣ(−sin x) = 2e²ˣ cos x − e²ˣ sin x. This can be factored as e²ˣ(2cos x − sin x). Note how the chain rule was essential inside the product rule to correctly differentiate e²ˣ.

Question 4: APPLIED

The height of a wave on a beach is modelled by h(t) = (t + 1) · sin(πt/6), where h is in metres and t is in seconds. Find the rate of change of height at t = 3 seconds.

Show answer

We need h′(3). Set u = t + 1 and v = sin(πt/6). Then u′ = 1 and, by the chain rule, v′ = cos(πt/6) · π/6. Product rule: h′(t) = (1) · sin(πt/6) + (t + 1) · cos(πt/6) · π/6. At t = 3: sin(π·3/6) = sin(π/2) = 1 and cos(π/2) = 0. So h′(3) = 1 · 1 + (4) · 0 · π/6 = 1. The rate of change of height at t = 3 seconds is 1 metre per second. The cosine term vanishes because at this instant the trig argument equals π/2, illustrating how specific time values can simplify results dramatically.

Question 5: CRITICAL THINKING

Let f(x) = ln(sin x) for 0 < x < π. Use the chain rule to find f′(x), then determine the value of x in the interval (0, π) where f′(x) = 0. Explain what this tells you about f(x) at that point.

Show answer

Using the chain rule with outer function ln(·) and inner function sin x: f′(x) = (1/sin x) · cos x = cos x / sin x = cot x. Setting f′(x) = 0: cot x = 0, which means cos x = 0. In the interval (0, π), this occurs at x = π/2. At this point, f(x) = ln(sin(π/2)) = ln(1) = 0. Since f′(π/2) = 0 and we can verify f″(π/2) < 0, this is a local maximum. This means the natural log of sin x achieves its greatest value of 0 when sin x reaches its maximum of 1. For all other x in (0, π), sin x < 1, so ln(sin x) < 0, confirming the maximum.

SUMMARY

Lesson Summary

Summary

IB Math AA SL 5.3 equips you with a complete toolkit for differentiation. The power rule (d/dx [xⁿ] = nxⁿ⁻¹) handles polynomials and fractional or negative exponents. The trigonometric rules give d/dx [sin x] = cos x and d/dx [cos x] = −sin x. The exponential rule (d/dx [eˣ] = eˣ) and the logarithmic rule (d/dx [ln x] = 1/x) complete the set of basic derivatives.

For combinations, the product rule (u′v + uv′) handles multiplied functions, the quotient rule ((u′v − uv′)/v²) handles divided functions, and the chain rule (f′(g(x)) · g′(x)) handles composite functions. Most IB exam questions require combining two or more of these rules. The key to success is recognising the structure of the function first, selecting the correct rule, and then carefully carrying out each step — especially remembering to multiply by the inner derivative when using the chain rule.