04-04 - Differentiation Rules
Phase: 4 | Subject: 04-04 Prerequisites: 04-03-the-derivative.md (limit definition) Next subject: 04-05-derivatives-of-elementary-functions.md
Learning Objectives
By the end of this subject, you will be able to:
- Apply the power rule for polynomial functions
- Apply the constant multiple rule
- Apply the sum/difference rule
- Apply the product rule
- Apply the quotient rule
- Apply the chain rule for composite functions
Core Content
Power Rule
$d/dx[xⁿ] = n·x^(n-1)$
Example: d/dx[x⁵] = 5x⁴ Example: d/dx[x³] = 3x² Example: d/dx[x] = 1 (since x = x¹, derivative = 1·x⁰ = 1) Example: d/dx[1] = 0 (constant)
Constant Multiple Rule
$d/dx[k·f(x)] = k·f'(x)$
Example: d/dx[3x⁴] = 3 · 4x³ = 12x³
Sum/Difference Rule
$d/dx[f(x) ± g(x)] = f'(x) ± g'(x)$
Example: d/dx[x³ + 2x² - 5x + 1] = 3x² + 4x - 5
Product Rule
$d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)$
Mnemonic: "First times derivative of second, plus second times derivative of first."
Example: d/dx[(2x + 1)(3x² - 1)] f = 2x + 1, f' = 2. g = 3x² - 1, g' = 6x. f'(x) = 2(3x² - 1) + (2x + 1)(6x) = 6x² - 2 + 12x² + 6x = 18x² + 6x - 2
Quotient Rule
$d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²$
Mnemonic: "Low d-High minus High d-Low, over Low squared."
Example: d/dx[(x² + 1)/(x + 2)] f = x² + 1, f' = 2x. g = x + 2, g' = 1. = [2x(x + 2) - (x² + 1)(1)] / (x + 2)² = [2x² + 4x - x² - 1] / (x + 2)² = [x² + 4x - 1] / (x + 2)²
Chain Rule
⚠️ THIS IS CRITICAL — the chain rule is the most-used differentiation technique. You will use it constantly: in implicit differentiation, related rates, integration by substitution, multivariable calculus, and backpropagation in neural networks.
For composite functions f(g(x)):
$d/dx[f(g(x))] = f'(g(x)) · g'(x)$
Mnemonic: "Derivative of the outside, leave the inside alone, times derivative of the inside."
Example: d/dx[(2x + 3)⁵] Outside: u⁵, derivative = 5u⁴ Inside: 2x + 3, derivative = 2 Result: 5(2x + 3)⁴ · 2 = 10(2x + 3)⁴
Example: d/dx[√(x² + 1)] = d/dx[(x² + 1)^(1/2)] Outside: u^(1/2), derivative = (1/2)u^(-1/2) Inside: x² + 1, derivative = 2x Result: (1/2)(x² + 1)^(-1/2) · 2x = x/√(x² + 1)
Key Terms
- 04 04 Differentiation Rules
- Chain Rule
- Constant Multiple Rule
- Correct: A)
- Correct: B)
- Correct: D)
- Example 1: Power rule and sum/difference
- Example 2: Product rule
- Example 3: Chain rule
- Power Rule
- Product Rule
- Quotient Rule
Worked Examples
Example 1: Power rule and sum/difference
f(x) = 5x⁴ - 3x² + 7x - 2 f'(x) = 20x³ - 6x + 7
Example 2: Product rule
f(x) = (x + 1)(x² - 2) f'(x) = (1)(x² - 2) + (x + 1)(2x) = x² - 2 + 2x² + 2x = 3x² + 2x - 2
Example 3: Chain rule
f(x) = √(3x + 1) = (3x + 1)^(1/2) Outside: u^(1/2), derivative = (1/2)u^(-1/2) Inside: 3x + 1, derivative = 3 f'(x) = (1/2)(3x + 1)^(-1/2) · 3 = 3/(2√(3x + 1))
Quiz
Q1: What does the concept of Chain Rule primarily refer to in this subject?
A) A historical anecdote about Chain Rule B) The definition and application of Chain Rule C) A computational error related to Chain Rule D) A visual representation of Chain Rule
Correct: B)
- If you chose A: This is incorrect. Chain Rule is defined as: the definition and application of chain rule. The other options describe different aspects that are not the primary focus.
- If you chose B: Chain Rule is defined as: the definition and application of chain rule. The other options describe different aspects that are not the primary focus. Correct!
- If you chose C: This is incorrect. Chain Rule is defined as: the definition and application of chain rule. The other options describe different aspects that are not the primary focus.
- If you chose D: This is incorrect. Chain Rule is defined as: the definition and application of chain rule. The other options describe different aspects that are not the primary focus.
Q2: Which of the following is the key formula discussed in this subject?
A) An unrelated formula from a different topic B) d/dx[xⁿ] = n·x^(n-1) C) A simplified version of d/dx[xⁿ] = n·x^(n-1)... D) The inverse operation of the formula in question
Correct: B)
- If you chose A: This is incorrect. The formula d/dx[xⁿ] = n·x^(n-1) is central to this subject. The other options are either simplified versions or unrelated.
- If you chose B: The formula d/dx[xⁿ] = n·x^(n-1) is central to this subject. The other options are either simplified versions or unrelated. Correct!
- If you chose C: This is incorrect. The formula d/dx[xⁿ] = n·x^(n-1) is central to this subject. The other options are either simplified versions or unrelated.
- If you chose D: This is incorrect. The formula d/dx[xⁿ] = n·x^(n-1) is central to this subject. The other options are either simplified versions or unrelated.
Q3: What is the primary purpose of Constant Multiple Rule?
A) It is used only in advanced research contexts B) It replaces all other methods in this domain C) It is used to constant multiple rule in mathematical analysis D) It is primarily a historical notation system
Correct: C)
- If you chose A: This is incorrect. Constant Multiple Rule serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose B: This is incorrect. Constant Multiple Rule serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose C: Constant Multiple Rule serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
- If you chose D: This is incorrect. Constant Multiple Rule serves the purpose described in the correct answer. The other options misrepresent its role.
Q4: Which statement about Power Rule is TRUE?
A) Power Rule is a fundamental concept covered in this subject B) Power Rule is an advanced topic beyond this subject's scope C) Power Rule is not related to this subject D) Power Rule is mentioned only as a historical footnote
Correct: A)
- If you chose A: Power Rule is a fundamental concept covered in this subject. This subject covers Power Rule as part of its core content. Correct!
- If you chose B: This is incorrect. Power Rule is a fundamental concept covered in this subject. This subject covers Power Rule as part of its core content.
- If you chose C: This is incorrect. Power Rule is a fundamental concept covered in this subject. This subject covers Power Rule as part of its core content.
- If you chose D: This is incorrect. Power Rule is a fundamental concept covered in this subject. This subject covers Power Rule as part of its core content.
Q5: Based on the worked examples in this subject, what is the correct result?
A) 10(2x + 3)⁴ B) An unrelated numerical value C) The inverse of the correct answer D) A different result from a common mistake
Correct: A)
- If you chose A: The worked examples show that the result is 10(2x + 3)⁴. The other options represent common errors. Correct!
- If you chose B: This is incorrect. The worked examples show that the result is 10(2x + 3)⁴. The other options represent common errors.
- If you chose C: This is incorrect. The worked examples show that the result is 10(2x + 3)⁴. The other options represent common errors.
- If you chose D: This is incorrect. The worked examples show that the result is 10(2x + 3)⁴. The other options represent common errors.
Q6: How are Power Rule and Product Rule related?
A) Power Rule and Product Rule are closely related concepts B) Power Rule is the inverse of Product Rule C) Power Rule is a special case of Product Rule D) Power Rule and Product Rule are completely unrelated topics
Correct: A)
- If you chose A: Both Power Rule and Product Rule are covered in this subject as interconnected topics. Correct!
- If you chose B: This is incorrect. Both Power Rule and Product Rule are covered in this subject as interconnected topics.
- If you chose C: This is incorrect. Both Power Rule and Product Rule are covered in this subject as interconnected topics.
- If you chose D: This is incorrect. Both Power Rule and Product Rule are covered in this subject as interconnected topics.
Q7: What is a common pitfall when working with Quotient Rule?
A) Quotient Rule has no common misconceptions B) A common mistake is confusing Quotient Rule with a similar concept C) The main error with Quotient Rule is using it when it is not needed D) Quotient Rule is always computed the same way in all contexts
Correct: B)
- If you chose A: This is incorrect. Students often confuse Quotient Rule with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose B: Students often confuse Quotient Rule with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
- If you chose C: This is incorrect. Students often confuse Quotient Rule with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose D: This is incorrect. Students often confuse Quotient Rule with similar-sounding or related concepts. Pay attention to the precise definitions.
Q8: When should you apply Sum/Difference Rule?
A) Apply Sum/Difference Rule to solve problems in this subject's domain B) Use Sum/Difference Rule only in pure mathematics contexts C) Avoid Sum/Difference Rule unless explicitly instructed D) Sum/Difference Rule is not practically useful
Correct: A)
- If you chose A: Sum/Difference Rule is a practical tool used throughout this subject to solve relevant problems. Correct!
- If you chose B: This is incorrect. Sum/Difference Rule is a practical tool used throughout this subject to solve relevant problems.
- If you chose C: This is incorrect. Sum/Difference Rule is a practical tool used throughout this subject to solve relevant problems.
- If you chose D: This is incorrect. Sum/Difference Rule is a practical tool used throughout this subject to solve relevant problems.
Practice Problems
-
d/dx[7x⁶] = ? Answer: 42x⁵
-
d/dx[x⁵ + 3x³ - x] = ? Answer: 5x⁴ + 9x² - 1
-
d/dx[(x² + 1)(x³ - 2)] = ? Answer: Product rule: 2x(x³ - 2) + (x² + 1)(3x²) = 2x⁴ - 4x + 3x⁴ + 3x² = 5x⁴ + 3x² - 4x
-
d/dx[(x² + 1)/(x - 1)] = ? Answer: Quotient rule: [2x(x - 1) - (x² + 1)(1)] / (x - 1)² = [2x² - 2x - x² - 1] / (x - 1)² = (x² - 2x - 1)/(x - 1)²
-
d/dx[(x² + 4x)^(5)] = ? Answer: Chain rule: 5(x² + 4x)⁴ · (2x + 4) = 10(x + 2)(x² + 4x)⁴
Summary
Key takeaways:
- Power rule: d/dx[xⁿ] = n·x^(n-1)
- Constant multiple: pull out constants
- Sum/difference: differentiate term by term
- Product: f'g + fg'
- Quotient: (f'g - fg')/g²
- Chain: f'(g(x)) · g'(x) — outside derivative times inside derivative
Pitfalls
- Forgetting the chain rule (missing the inner derivative): This is the single most common differentiation error. When differentiating a composite function like (2x + 3)⁵, you must multiply by the derivative of the INSIDE. Writing just 5(2x + 3)⁴ is wrong — the correct answer is 5(2x + 3)⁴ · 2 = 10(2x + 3)⁴. Always ask: "Did I multiply by the derivative of the inside?"
- Confusing the product and quotient rules: The product rule is f'g + fg' (plus sign). The quotient rule is (f'g - fg')/g² (minus sign in numerator, denominator squared). Students often swap the signs or forget to square the denominator in the quotient rule.
- Power rule index errors: d/dx[xⁿ] = n·x^(n-1) — multiply by n and REDUCE the exponent by 1. Common mistakes: forgetting to subtract 1 (writing n·xⁿ), or reducing the coefficient instead of the exponent. Also remember that d/dx[x] = 1 and d/dx[c] = 0 for constants.
- Applying the product rule when a simpler rule suffices: For something like 3(x² + 1), use the constant multiple rule: d/dx = 3·2x = 6x. Using the product rule (3 as a function, x²+1 as another) works but adds unnecessary work and risk of error. Recognise when constant multiple or sum/difference rules are sufficient.
- Misordering terms in the quotient rule numerator: The numerator is f'g - fg', NOT g'f - gf' or fg' - f'g. The mnemonic "low d-high minus high d-low" helps: denominator times derivative of numerator, minus numerator times derivative of denominator. Getting the subtraction order wrong flips the sign of the entire derivative.