UK K12 GCSE/A-Level: Year 12 A-Level Lower Sixth Mathematics - Pure Differentiation, Chain, Product, Quotient Rules

Learning Objectives

By the end of this topic, students will be able to:

• Apply the chain rule to differentiate composite functions
• Apply the product rule to differentiate functions of the form u(x)v(x)
• Apply the quotient rule to differentiate functions of the form u(x)/v(x)
• Recognize and apply the appropriate rule for differentiation in a given context
• Evaluate the derivative of a function using the chain, product, and quotient rules

Core Concepts

The chain rule is used to differentiate composite functions of the form f(g(x)). It states that if y = f(u) and u = g(x), then y' = f'(u) * g'(x). This can be visualized as a chain of functions, where the derivative of the outer function is multiplied by the derivative of the inner function.

The product rule is used to differentiate functions of the form u(x)v(x). It states that if y = u(x)v(x), then y' = u'(x)v(x) + u(x)v'(x). This can be thought of as the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.

The quotient rule is used to differentiate functions of the form u(x)/v(x). It states that if y = u(x)/v(x), then y' = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2. This can be thought of as the derivative of the numerator multiplied by the denominator, minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator.

Worked Examples

Example 1: Chain Rule

Find the derivative of y = (2x + 1)^3.

Using the chain rule, we can write y = f(u) where u = 2x + 1. Then, y' = f'(u) * g'(x) = 3(2x + 1)^2 * 2.

Example 2: Product Rule

Find the derivative of y = x^2 * sin(x).

Using the product rule, we can write y = u(x)v(x) where u(x) = x^2 and v(x) = sin(x). Then, y' = u'(x)v(x) + u(x)v'(x) = 2x * sin(x) + x^2 * cos(x).

Example 3: Quotient Rule

Find the derivative of y = (x^2 + 1) / (x + 1).

Using the quotient rule, we can write y = u(x)/v(x) where u(x) = x^2 + 1 and v(x) = x + 1. Then, y' = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2 = ((2x)(x + 1) - (x^2 + 1)(1)) / (x + 1)^2.

Common Misconceptions

• Students often confuse the chain rule with the product rule, and vice versa. Make sure to emphasize the key difference between the two rules.
• Students may struggle to apply the quotient rule correctly, especially when dealing with complex fractions. Encourage them to break down the problem step-by-step and to check their work.
• Students may assume that the derivative of a function is always positive or always negative. Remind them that the derivative can change sign depending on the function and the point at which it is evaluated.

Exam Tips

• Make sure to read the question carefully and understand what is being asked.
• Identify the type of function being differentiated and choose the appropriate rule.
• Break down complex problems into smaller, more manageable parts.
• Check your work and make sure that your answer is reasonable.

MCQs

Question 1: [F]

What is the derivative of y = (2x + 1)^3?

A) 6(2x + 1)^2 B) 3(2x + 1)^2 C) 2(2x + 1)^2 D) (2x + 1)^3

Correct answer: B) 3(2x + 1)^2 Why the distractors fail: A) is incorrect because it is the derivative of the outer function, but not the inner function. C) is incorrect because it is the derivative of the inner function, but not the outer function. D) is incorrect because it is the original function, not its derivative.

Question 2: [H]

What is the derivative of y = x^2 * sin(x)?

A) 2x * sin(x) - x^2 * cos(x) B) 2x * sin(x) + x^2 * cos(x) C) x^2 * cos(x) - 2x * sin(x) D) x^2 * sin(x) - 2x * cos(x)

Correct answer: B) 2x * sin(x) + x^2 * cos(x) Why the distractors fail: A) is incorrect because it is the derivative of the first function multiplied by the second function, but not the derivative of the second function. C) is incorrect because it is the derivative of the second function, but not the first function. D) is incorrect because it is the original function, not its derivative.

Question 3: [F]

What is the derivative of y = (x^2 + 1) / (x + 1)?

A) ((2x)(x + 1) - (x^2 + 1)(1)) / (x + 1)^2 B) ((x^2 + 1)(1) - (2x)(x + 1)) / (x + 1)^2 C) ((x^2 + 1)(1) + (2x)(x + 1)) / (x + 1)^2 D) ((2x)(x + 1) + (x^2 + 1)(1)) / (x + 1)^2

Correct answer: A) ((2x)(x + 1) - (x^2 + 1)(1)) / (x + 1)^2 Why the distractors fail: B) is incorrect because it has the wrong sign. C) is incorrect because it has the wrong numerator. D) is incorrect because it has the wrong denominator.

Question 4: [H]

What is the derivative of y = x^3 * sin(x)?

A) 3x^2 * sin(x) + x^3 * cos(x) B) 3x^2 * sin(x) - x^3 * cos(x) C) x^3 * cos(x) - 3x^2 * sin(x) D) x^3 * sin(x) - 3x^2 * cos(x)

Correct answer: A) 3x^2 * sin(x) + x^3 * cos(x) Why the distractors fail: B) is incorrect because it has the wrong sign. C) is incorrect because it is the derivative of the second function, but not the first function. D) is incorrect because it is the original function, not its derivative.

Question 5: [F]

What is the derivative of y = (2x + 1)^2?

A) 4(2x + 1) B) 2(2x + 1) C) (2x + 1)^2 D) 2(2x + 1)^2

Correct answer: B) 2(2x + 1) Why the distractors fail: A) is incorrect because it is the derivative of the outer function, but not the inner function. C) is incorrect because it is the original function, not its derivative. D) is incorrect because it is the derivative of the outer function, but not the inner function.

Short-answer questions

1. Find the derivative of y = (x^2 + 1)^3.
2. Find the derivative of y = x^2 * sin(x) using the product rule.
3. Find the derivative of y = (x^2 + 1) / (x + 1) using the quotient rule.
4. Find the derivative of y = x^3 * sin(x) using the product rule.
5. Find the derivative of y = (2x + 1)^2.