Differentiation Rules — SL 5.3 Differentiation rules (power; common functions) (as included) - IB Mathematics Applications and Interpretation

SECTION 1

Historical Context & Motivation

Before calculus existed, mathematicians could only approximate how quickly things changed. If you wanted to know how fast a cannonball was moving at a specific instant, you had to measure its position at two nearby times and divide the distance by the time elapsed — essentially drawing a secant line and hoping for the best. This approach was clumsy and imprecise, and scientists knew there had to be a better way.

In the seventeenth century, two brilliant minds independently developed the mathematical machinery to solve this problem. Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany each created what we now call differential calculus — a set of tools for finding exact instantaneous rates of change. Over the centuries that followed, mathematicians refined these ideas into elegant, efficient rules that let us skip the long limit process entirely.

1665 • Newton's Method of Fluxions

Isaac Newton develops his "method of fluxions" during the Great Plague, describing rates of change as flowing quantities. He discovers the power rule as one of his first results.

1684 • Leibniz Publishes His Calculus

Leibniz publishes the first paper on differential calculus, introducing the notation dy/dx that we still use today. His systematic approach makes differentiation rules accessible to a wider audience.

1748 • Euler's Analysis of Functions

Leonhard Euler publishes Introductio in analysin infinitorum, formalising the differentiation of exponential and trigonometric functions and establishing the rules we study in SL 5.3.

1821 • Cauchy Rigourises Calculus

Augustin-Louis Cauchy places the differentiation rules on a firm logical foundation using limits, giving us the rigorous definitions that underpin every shortcut rule.

The central question these mathematicians answered is one you will master in this lesson: Given a function, how do you efficiently find its derivative — the function that tells you the instantaneous rate of change at every point? The differentiation rules of SL 5.3 are the direct descendants of Newton's and Leibniz's groundbreaking work, packaged into formulas you can apply in seconds.

SECTION 2

Core Principles & Definitions

Before diving into specific rules, you need to understand a few foundational ideas. The derivative of a function f(x), written as f′(x) or dy/dx, tells you the slope of the tangent line to the curve at any point x. In real-world terms, the derivative captures how fast an output changes when you nudge the input. The differentiation rules you'll learn here are shortcuts that spare you from computing limits every single time.

Power Rule

If f(x) = xⁿ, then f′(x) = nxⁿ⁻¹. Bring the exponent down as a coefficient, then reduce the exponent by one.

Constant Multiple Rule

If f(x) = c · g(x), then f′(x) = c · g′(x). A constant factor stays put — you only differentiate the variable part.

Sum / Difference Rule

The derivative of a sum (or difference) equals the sum (or difference) of the derivatives. Differentiate each term separately.

Exponential Rule

If f(x) = e^x, then f′(x) = e^x. The natural exponential function is its own derivative — a remarkable property.

Trigonometric Rules

The derivative of sin x is cos x, and the derivative of cos x is −sin x. These two functions cycle into each other under differentiation.

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KEY TAKEAWAY

Think of differentiation rules as a speed-dial for slopes. Instead of going through the full limit definition every time — like calculating a tip from scratch with long division — the rules give you a quick formula. The power rule, for instance, is like knowing that a 20 % tip on $50 is $10 without pulling out a calculator. You get the exact answer, faster.

SECTION 3

Visual Explanation — The Power Rule in Action

The diagram below shows the function f(x) = x² alongside its derivative f′(x) = 2x. Notice how wherever the parabola is steepest, the derivative has the largest absolute value, and wherever the parabola is flat (at the vertex), the derivative crosses zero. This visual connection is at the heart of differentiation: the derivative graph is essentially a slope tracker for the original curve.

The cyan parabola represents f(x) = x². The violet line is its derivative f′(x) = 2x. At the vertex (x = 0), the parabola is flat, so the derivative equals zero. The dashed amber line shows the tangent at x = 1, where the slope is 2 — matching f′(1) = 2(1) = 2.

This relationship is universal: whenever the original function rises steeply, its derivative is a large positive number. Whenever the function falls steeply, the derivative is a large negative number. And at any turning point — a peak or valley — the derivative passes through zero. Keep this visual intuition in mind as you learn each rule; it will help you check whether your differentiated answer makes sense.

SECTION 4

Mathematical Framework — The Rules

Below are the key differentiation rules covered in IB AI SL 5.3. Each rule is stated as a formula, followed by a brief note explaining the variables and when to use it. In your exam formula booklet you will find most of these listed, but understanding why they work will help you apply them confidently rather than just memorising them.

POWER RULE

f(x) = xⁿ ⟹ f′(x) = n·xⁿ⁻¹

n can be any real number — positive integers, negative integers, or fractions. For example, √x = x^1/2 so its derivative is (1/2)x^−1/2.

CONSTANT RULE

f(x) = c ⟹ f′(x) = 0

The derivative of any constant is zero, because a horizontal line has no slope.

CONSTANT MULTIPLE & SUM/DIFFERENCE RULES

d/dx [c·f(x)] = c·f′(x) and d/dx [f(x) ± g(x)] = f′(x) ± g′(x)

These rules let you handle polynomials term by term. Factor out constants, then differentiate each term independently and add or subtract the results.

NATURAL EXPONENTIAL RULE

f(x) = eˣ ⟹ f′(x) = eˣ

The function e^x is unique: it is the only function that equals its own derivative. Here e ≈ 2.718 is Euler's number.

TRIGONOMETRIC RULES

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

These apply when x is measured in radians (not degrees). Remember the negative sign on the cosine derivative — it trips up many students.

NATURAL LOGARITHM RULE

f(x) = ln x ⟹ f′(x) = 1/x (x > 0)

The derivative of the natural logarithm is the reciprocal function 1/x, defined only for positive x.

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IB Exam Tip

Your formula booklet lists these rules. In the exam, you don't need to derive them from first principles — but you do need to recognise which rule to apply and combine them correctly. Practice rewriting expressions (like 1/x² = x⁻²) before differentiating.

SECTION 5

Detailed Breakdown — Derivative Reference Chart

The table below collects all the SL 5.3 differentiation rules in one place, alongside a concrete example for each. Use it as a quick-reference sheet when working through problems. After the table, a diagram shows how the derivatives of the trigonometric and exponential functions behave.

Function f(x)	Derivative f′(x)	Example
c (constant)	0	f(x) = 7 → f′(x) = 0
xⁿ	n·xⁿ⁻¹	f(x) = x⁵ → f′(x) = 5x⁴
c·xⁿ	c·n·xⁿ⁻¹	f(x) = 3x⁴ → f′(x) = 12x³
eˣ	eˣ	f(x) = eˣ → f′(x) = eˣ
ln x	1/x	f(x) = ln x → f′(x) = 1/x
sin x	cos x	f(x) = sin x → f′(x) = cos x
cos x	−sin x	f(x) = cos x → f′(x) = −sin x
x⁻ⁿ (i.e. 1/xⁿ)	−n·x⁻ⁿ⁻¹	f(x) = x⁻² → f′(x) = −2x⁻³

Complete SL 5.3 Differentiation Rules at a Glance

The sin x curve (solid cyan) has its derivative shown as the dashed violet cos x curve. Differentiating cos x yields −sin x (dotted pink). The inset panel shows that eˣ is its own derivative.

There are two important observations to take from the diagram. First, the trigonometric derivatives form a four-step cycle: sin → cos → −sin → −cos → sin. Each differentiation shifts the curve to the left by π/2. Second, the exponential curve eˣ never changes shape under differentiation — it is the only elementary function with this self-replicating property.

SECTION 6

Worked Example — Differentiating a Mixed Function

Let's differentiate a function that combines several of the SL 5.3 rules. Find f′(x) for:

GIVEN FUNCTION

f(x) = 4x³ − 2x² + 5x − 7 + 3eˣ + sin x

This function has polynomial terms, an exponential term, and a trigonometric term — a perfect showcase of the sum rule combined with individual rules.

Differentiating f(x) = 4x³ − 2x² + 5x − 7 + 3eˣ + sin x

1
Step 1 — Apply the Sum/Difference Rule

By the sum/difference rule, we can differentiate each term separately. Write: f′(x) = d/dx(4x³) − d/dx(2x²) + d/dx(5x) − d/dx(7) + d/dx(3eˣ) + d/dx(sin x).
2
Step 2 — Differentiate the Polynomial Terms (Power Rule)

Apply the power rule to each polynomial term. For 4x³: bring down the 3, reduce the exponent by 1 → 4 × 3 × x² = 12x². For 2x²: → 2 × 2 × x¹ = 4x. For 5x (which is 5x¹): → 5 × 1 × x⁰ = 5. For the constant −7: the derivative is 0.

Polynomial part: 12x² − 4x + 5
3
Step 3 — Differentiate the Exponential Term

The constant multiple rule says we keep the 3 and differentiate eˣ. Since d/dx(eˣ) = eˣ, we get 3eˣ.

Exponential part: 3eˣ
4
Step 4 — Differentiate the Trigonometric Term

The derivative of sin x is cos x. So d/dx(sin x) = cos x.

Trigonometric part: cos x
5
Step 5 — Combine All Parts

Put all the differentiated terms back together.

f′(x) = 12x² − 4x + 5 + 3eˣ + cos x

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Quick Check

You can verify your answer by substituting a value. At x = 0: f(0) = 0 − 0 + 0 − 7 + 3(1) + 0 = −4. f′(0) = 0 − 0 + 5 + 3(1) + 1 = 9. This tells us the curve is rising at x = 0, with a slope of 9. You could sketch this to confirm it looks reasonable.

SECTION 7

Strengths, Limitations & Common Mistakes

The SL 5.3 differentiation rules are powerful but they have a defined scope. Understanding what these rules can and cannot do will save you from errors on exams and prepare you for the additional rules you'll encounter later.

Strengths	Limitations
Handle any polynomial instantly — no matter how many terms or how large the exponents.	Cannot directly differentiate products like x² · sin x (need the product rule, covered in HL).
Work for negative and fractional exponents, allowing you to differentiate functions like 1/x or √x.	Cannot handle compositions like sin(3x) or e²ˣ without the chain rule (also HL).
Cover the most common function families: polynomials, exponentials, logarithms, and basic trig.	Do not apply to functions like tan x, aˣ (where a ≠ e), or inverse trig without additional formulas.
Can be combined using the sum/difference and constant multiple rules to handle complex expressions.	Require expressions to be in the correct form first — e.g., rewrite 1/x³ as x⁻³ before applying the power rule.

What the SL 5.3 Rules Can and Cannot Do

Common Mistakes to Avoid

✓ Forgetting to subtract 1 from the exponent. If f(x) = x⁴, the derivative is 4x³, not 4x⁴. The exponent must decrease.
✓ Leaving the constant term in the derivative. The derivative of 12 is 0, not 12. Constants vanish under differentiation.
✓ Sign error on cos x. The derivative of cos x is −sin x. Many students drop the negative sign.
✓ Using degrees instead of radians. Make sure your calculator is in radian mode when evaluating trig derivatives.

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KEY TAKEAWAY

The SL 5.3 rules are like a toolbox with specific tools — a screwdriver (power rule), a wrench (exponential rule), and pliers (trig rules). Each tool works perfectly for its intended job. But if you encounter a bolt that's inside another bolt (a composite function) or two parts welded together (a product), you'll need additional tools from the HL toolkit. For now, learn to recognise which tool fits each problem.

SECTION 8

Connection to Advanced Differentiation

The rules you've learned form the foundation for more advanced techniques. If you continue to HL or study calculus at university, you will encounter rules that build directly on top of the SL 5.3 toolkit. Understanding the connections now will make those transitions smoother.

SL 5.3 Rule (What You Know)	Advanced Extension (What Comes Next)	Example
d/dx(xⁿ) = nxⁿ⁻¹	Chain Rule: d/dx[(g(x))ⁿ] = n·[g(x)]ⁿ⁻¹ · g′(x)	d/dx[(2x + 1)³] = 3(2x + 1)² · 2
d/dx(eˣ) = eˣ	Chain Rule: d/dx( $e^{g(x)}$ ) = $e^{g(x)}$ · g′(x)	d/dx(e³ˣ) = 3e³ˣ
d/dx(sin x) = cos x	Chain Rule: d/dx(sin(g(x))) = cos(g(x)) · g′(x)	d/dx(sin 2x) = 2cos 2x
Sum/difference rule only	Product Rule: d/dx[f·g] = f′g + fg′	d/dx(x²·sin x) = 2x·sin x + x²·cos x
d/dx(ln x) = 1/x	Chain Rule: d/dx(ln(g(x))) = g′(x)/g(x)	d/dx(ln(x²)) = 2x/x² = 2/x

SL 5.3 as a Foundation for Advanced Rules

Notice a pattern: the chain rule is the key upgrade. It extends every SL rule to handle composite functions. The good news is that you already know the "inner" part — the SL rule tells you the derivative of the outer function. The chain rule simply multiplies by the derivative of what's inside. Mastering SL 5.3 thoroughly means you're already halfway there.

SECTION 9

Practice Problems

Work through these five problems in order. They start simple and build in complexity. Try each one on paper before revealing the answer.

Question 1: CONCEPTUAL

Explain in your own words why the derivative of a constant (such as f(x) = 9) is zero. What does this mean graphically?

Show answer

A constant function produces a horizontal line on the graph. Since the slope of a horizontal line is zero everywhere, the derivative must be zero. In terms of rates of change, if a quantity never changes (it's always 9, for example), then its rate of change is zero at every point. This is consistent with the power rule: writing 9 as 9x⁰, the derivative is 9 × 0 × x⁻¹ = 0.

Question 2: BASIC CALCULATION

Find f′(x) if f(x) = 6x⁵ − 3x² + 8x − 11.

Show answer

Apply the power rule term by term. For 6x⁵: 6 × 5 × x⁴ = 30x⁴. For −3x²: −3 × 2 × x¹ = −6x. For 8x: 8 × 1 × x⁰ = 8. For −11: 0. Combining: f′(x) = 30x⁴ − 6x + 8. Each term was differentiated independently using the sum rule and the constant multiple rule.

Question 3: INTERMEDIATE

Differentiate g(x) = 4/x² + 3√x − 2eˣ. (Hint: rewrite 4/x² and 3√x using negative and fractional exponents first.)

Show answer

Rewrite the function: g(x) = 4x⁻² +

3x^{1/2}

− 2eˣ. Now apply the rules. For 4x⁻²: 4 × (−2) × x⁻³ = −8x⁻³ = −8/x³. For

3x^{1/2}

: 3 × (1/2) ×

x^{−1/2}

= (3/2)

x^{−1/2}

= 3/(2√x). For −2eˣ: −2eˣ (since d/dx(eˣ) = eˣ). Therefore g′(x) = −8/x³ + 3/(2√x) − 2eˣ. The key step was rewriting the original terms into power-rule-friendly form before differentiating.

Question 4: APPLIED

The height h (in metres) of a ball thrown upward is modelled by h(t) = −4.9t² + 20t + 1.5, where t is time in seconds. Find the velocity function h′(t) and determine the exact time when the ball reaches its maximum height.

Show answer

The velocity is the derivative of height with respect to time. Applying the power rule: h′(t) = −4.9 × 2 × t + 20 × 1 + 0 = −9.8t + 20. At maximum height, the velocity equals zero (the ball momentarily stops rising). Setting −9.8t + 20 = 0 gives t = 20/9.8 ≈ 2.04 seconds. This means the ball reaches its peak at approximately 2.04 seconds after being thrown. The negative coefficient −9.8 represents gravitational acceleration (in m/s²), pulling the ball back down.

Question 5: CRITICAL THINKING

Consider the function f(x) = x³ − 6x² + 9x + 2. Find f′(x), then determine all values of x where the tangent line to the curve is horizontal. What does this tell you about the graph of f?

Show answer

Differentiating: f′(x) = 3x² − 12x + 9. A horizontal tangent means f′(x) = 0, so 3x² − 12x + 9 = 0. Dividing by 3: x² − 4x + 3 = 0. Factoring: (x − 1)(x − 3) = 0, so x = 1 or x = 3. At x = 1: f(1) = 1 − 6 + 9 + 2 = 6. At x = 3: f(3) = 27 − 54 + 27 + 2 = 2. The points (1, 6) and (3, 2) are where the tangent is horizontal, meaning they are local extrema (turning points). Since f is a cubic with a positive leading coefficient, (1, 6) is a local maximum and (3, 2) is a local minimum. This synthesis connects differentiation to understanding the shape of a graph.

SUMMARY

Lesson Summary

Summary

The SL 5.3 differentiation rules provide efficient shortcuts for finding derivatives without resorting to the limit definition. The power rule (d/dx of xⁿ = nxⁿ⁻¹) handles all polynomial-type terms, including those with negative and fractional exponents. The constant multiple rule and sum/difference rule allow you to differentiate complex expressions term by term. The derivative of a constant is always zero.

Beyond polynomials, the natural exponential function eˣ is its own derivative — a unique and important property. The trigonometric rules tell us that d/dx(sin x) = cos x and d/dx(cos x) = −sin x (always in radians). The natural logarithm rule gives d/dx(ln x) = 1/x for x > 0. Mastering these rules equips you to find instantaneous rates of change, locate turning points, and analyse real-world models — and sets the stage for the chain rule, product rule, and quotient rule in more advanced study.