📐 Concept diagram

04-03 - The Derivative

Phase: 4 | Subject: 04-03 Prerequisites: 04-02-continuity.md, 04-01-limits.md Next subject: 04-04-differentiation-rules.md

Learning Objectives

By the end of this subject, you will be able to:

1. Understand the derivative as an instantaneous rate of change
2. Understand the derivative as the slope of the tangent line
3. Use derivative notation (dy/dx, f'(x), y')
4. Calculate derivatives using the limit definition
5. Understand differentiability vs continuity

Core Content

Two Interpretations of the Derivative

1. Slope of the Tangent Line

For a curve y = f(x), the derivative at x = a gives the slope of the tangent line at the point (a, f(a)).

2. Instantaneous Rate of Change

The derivative tells you how fast f(x) is changing at exactly one moment, not an average over an interval.

Example: If s(t) is position, then s'(t) is velocity (instantaneous speed with direction).

The Limit Definition

⚠️ THIS IS CRITICAL — the limit definition of the derivative is the core idea of calculus. Everything else derives from this. Memorise it and understand why it works.

$f'(x) = $lim(h→0)$ [f(x + h) - f(x)] / h
$

This is the slope of the secant line as h → 0 (the two points merge into one tangent point).

Using the Limit Definition

Example: Find f'(x) for f(x) = x² using the limit definition.

f'(x) = $lim(h→0)$ [(x + h)² - x²] / h = $lim(h→0)$ [x² + 2xh + h² - x²] / h = $lim(h→0)$ [2xh + h²] / h = $lim(h→0)$ [2x + h] = 2x

So f'(x) = 2x.

Verification: At x = 3, tangent slope = 6. The parabola y = x² is steeper at x = 3 than at x = 1 (where slope = 2). Makes sense!

Derivative Notation

Differentiability and Continuity

Theorem: If f is differentiable at x = a, then f is continuous at x = a.

Converse is FALSE: A function can be continuous at a point but not differentiable there.

Examples of non-differentiability: - Corner: f(x) = |x| at x = 0 - Cusp: f(x) = x^(2/3) at x = 0 - Vertical tangent: f(x) = x^(1/3) at x = 0

Key Terms

• 04 03 The Derivative
• Correct: A)
• Correct: B)
• Derivative Notation
• Differentiability and Continuity
• Example 1: Derivative from definition
• Example 2: Tangent line
• Example 3: Differentiability check
• Instantaneous Rate of Change
• Notation
• Slope of the Tangent Line
• The Limit Definition

Worked Examples

Example 1: Derivative from definition

Find f'(x) for f(x) = 3x² - 2x + 1.

f'(x) = $lim(h→0)$ [3(x+h)² - 2(x+h) + 1 - (3x² - 2x + 1)] / h = $lim(h→0)$ [3(x² + 2xh + h²) - 2x - 2h + 1 - 3x² + 2x - 1] / h = $lim(h→0)$ [3x² + 6xh + 3h² - 2x - 2h + 1 - 3x² + 2x - 1] / h = $lim(h→0)$ [6xh + 3h² - 2h] / h = $lim(h→0)$ [6x + 3h - 2] = 6x - 2

Example 2: Tangent line

Find the equation of the tangent to y = x² at x = 2.

1. f'(x) = 2x, so f'(2) = 4 (slope)
2. Point: (2, 4)
3. Equation: y - 4 = 4(x - 2)
4. y = 4x - 8 + 4 = 4x - 4

Example 3: Differentiability check

Is f(x) = |x| differentiable at x = 0?

Left derivative: $lim(h→0⁻)$ [|0 + h| - |0|] / h = $lim(h→0⁻)$ (-h)/h = -1 Right derivative: $lim(h→0⁺)$ [|0 + h| - |0|] / h = $lim(h→0⁺)$ h/h = 1

Since -1 ≠ 1, the derivative doesn't exist. Corner at x = 0.

Quiz

Q1: What does the concept of Derivative Notation primarily refer to in this subject?

A) The definition and application of Derivative Notation B) A historical anecdote about Derivative Notation C) A visual representation of Derivative Notation D) A computational error related to Derivative Notation

Correct: A)

• If you chose A: Derivative Notation is defined as: the definition and application of derivative notation. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Derivative Notation is defined as: the definition and application of derivative notation. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Derivative Notation is defined as: the definition and application of derivative notation. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Derivative Notation is defined as: the definition and application of derivative notation. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Differentiability and Continuity?

A) It is used to differentiability and continuity in mathematical analysis B) It is used only in advanced research contexts C) It replaces all other methods in this domain D) It is primarily a historical notation system

Correct: A)

• If you chose A: Differentiability and Continuity serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose B: This is incorrect. Differentiability and Continuity serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Differentiability and Continuity serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Differentiability and Continuity serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Instantaneous Rate of Change is TRUE?

A) Instantaneous Rate of Change is an advanced topic beyond this subject's scope B) Instantaneous Rate of Change is a fundamental concept covered in this subject C) Instantaneous Rate of Change is mentioned only as a historical footnote D) Instantaneous Rate of Change is not related to this subject

Correct: B)

• If you chose A: This is incorrect. Instantaneous Rate of Change is a fundamental concept covered in this subject. This subject covers Instantaneous Rate of Change as part of its core content.
• If you chose B: Instantaneous Rate of Change is a fundamental concept covered in this subject. This subject covers Instantaneous Rate of Change as part of its core content. Correct!
• If you chose C: This is incorrect. Instantaneous Rate of Change is a fundamental concept covered in this subject. This subject covers Instantaneous Rate of Change as part of its core content.
• If you chose D: This is incorrect. Instantaneous Rate of Change is a fundamental concept covered in this subject. This subject covers Instantaneous Rate of Change as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) 4. B) A different result from a common mistake C) An unrelated numerical value D) The inverse of the correct answer

Correct: A)

• If you chose A: The worked examples show that the result is 4.. The other options represent common errors. Correct!
• If you chose B: This is incorrect. The worked examples show that the result is 4.. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is 4.. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is 4.. The other options represent common errors.

Q5: How are Instantaneous Rate of Change and Notation related?

A) Instantaneous Rate of Change and Notation are completely unrelated topics B) Instantaneous Rate of Change is a special case of Notation C) Instantaneous Rate of Change and Notation are closely related concepts D) Instantaneous Rate of Change is the inverse of Notation

Correct: C)

• If you chose A: This is incorrect. Both Instantaneous Rate of Change and Notation are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Instantaneous Rate of Change and Notation are covered in this subject as interconnected topics.
• If you chose C: Both Instantaneous Rate of Change and Notation are covered in this subject as interconnected topics. Correct!
• If you chose D: This is incorrect. Both Instantaneous Rate of Change and Notation are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Slope of the Tangent Line?

A) The main error with Slope of the Tangent Line is using it when it is not needed B) Slope of the Tangent Line is always computed the same way in all contexts C) A common mistake is confusing Slope of the Tangent Line with a similar concept D) Slope of the Tangent Line has no common misconceptions

Correct: C)

• If you chose A: This is incorrect. Students often confuse Slope of the Tangent Line with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse Slope of the Tangent Line with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: Students often confuse Slope of the Tangent Line with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose D: This is incorrect. Students often confuse Slope of the Tangent Line with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply The Limit Definition?

A) Avoid The Limit Definition unless explicitly instructed B) Apply The Limit Definition to solve problems in this subject's domain C) The Limit Definition is not practically useful D) Use The Limit Definition only in pure mathematics contexts

Correct: B)

• If you chose A: This is incorrect. The Limit Definition is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: The Limit Definition is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose C: This is incorrect. The Limit Definition is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. The Limit Definition is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Use definition to find f'(x) for f(x) = 4x Answer: f'(x) = $lim(h→0)$ [4(x+h) - 4x]/h = $lim(h→0)$ 4h/h = 4. Constant slope = 4.

2. Find f'(x) for f(x) = x³ using definition Answer: f'(x) = 3x²

3. Tangent to y = x² at x = -1 Answer: Slope = -2, point (-1, 1). y - 1 = -2(x + 1), so y = -2x - 1.

4. Is f(x) = x^(1/3) differentiable at x = 0? Answer: f'(0) = $lim(h→0)$ h^(1/3)/h = $lim(h→0)$ 1/h^(2/3) = ∞. Vertical tangent — not differentiable.

5. If s(t) = 5t², what is the velocity at t = 3? Answer: v(t) = s'(t) = 10t. v(3) = 30.

Summary

Key takeaways:

• Derivative = instantaneous rate of change = slope of tangent
• Limit definition: f'(x) = $lim(h→0)$ [f(x+h) - f(x)]/h
• Differentiable implies continuous, but not conversely
• |x| is not differentiable at x = 0 (corner)
• f'(x) = 2x for f(x) = x²
• f'(x) = nx^(n-1) for f(x) = xⁿ (power rule, covered next)

Pitfalls

• Confusing the derivative with the original function: The derivative f'(x) is itself a NEW function — it gives the slope of f at each x, not the y-value. At x = 3 on f(x) = x², the y-value is 9, but the slope (derivative) is 6. Don't conflate function evaluation with derivative evaluation.
• Algebra errors in the difference quotient: Expanding (x + h)ⁿ correctly is essential. Common mistakes include: forgetting to square h (writing (x + h)² as x² + 2x + h² instead of x² + 2xh + h²), mishandling the subtraction, or incorrectly cancelling h. Work carefully through the algebra — these errors cascade.
• Assuming continuity implies differentiability: A function can be continuous but NOT differentiable (e.g., |x| at x = 0, has a corner). Differentiability is a stronger condition than continuity. Always check for corners, cusps, and vertical tangents when determining differentiability.
• Forgetting to check left and right derivatives at suspicious points: For piecewise functions or functions like |x|, compute the derivative from the left and right separately at the junction point. If the left derivative ≠ the right derivative (as with |x| at x = 0: -1 vs 1), the function is not differentiable there.
• Dropping the limit notation prematurely: The definition is f'(x) = $lim(h→0)$ [f(x+h) - f(x)]/h. Keep the "lim" notation until you've actually evaluated the limit. Writing just [f(x+h) - f(x)]/h without the limit is the difference quotient (average rate of change), not the derivative (instantaneous rate of change).

Next Steps

Next up: 04-04-differentiation-rules.md