Mathematics
Product Rule Practice Problems: Grade 12 Quiz
Use this product rule quiz to practice Grade 12 differentiation with 20 quick questions and build speed and accuracy. You will get instant results and simple steps along the way. Afterward, try the derivative rules quiz, explore the exponent rules quiz, or review polynomial practice problems.
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1What is the product rule formula for differentiation?
2Find the derivative of f(x) = x[@U00C2][@U00B7]sin(x) using the product rule.
3In the product rule formula f'(x)g(x) + f(x)g'(x), which term represents the derivative of the first fun<wbr>ction?
4What is the derivative of f(x) = 3x [@U00C2][@U00B7] x[@U00C2][@U00B2] using the product rule?
5Identify the mistake in the following differentiation: d/dx [u(x)v(x)] = u'(x)v'(x).
6Find the derivative of f(x) = x[@U00C2][@U00B2][@U00C2][@U00B7]cos(x) using the product rule.
7Differentiate f(x) = (3x + 1)[@U00C2][@U00B7]e[@U00CB][@U00A3] using the product rule.
8Compute the derivative of f(x) = x[@U00C2][@U00B7]ln(x) using the product rule.
9Find the derivative of f(x) = sin(x)[@U00C2][@U00B7]cos(x) using the product rule.
10Let f(x) = u(x)[@U00C2][@U00B7]v(x). If u(2) = 3, u'(2) = 4, v(2) = 5, and v'(2) = 6, what is f'(2)?
11Differentiate f(x) = (x[@U00C2][@U00B3] + 2x)(x - 1) using the product rule.
12Differentiate f(x) = e[@U00CB][@U00A3][@U00C2][@U00B7](x[@U00C2][@U00B2] - 1) using the product rule.
13Which of the following best explains the product rule's necessity in differentiation?
14Determine the derivative of f(x) = (5x - 2)[@U00C2][@U00B7][@U00E2][@U02C6][@U0161]x.
15Differentiate f(x) = (ln x)[@U00C2][@U00B7](x[@U00C2][@U00B2]) using the product rule.
16Differentiate f(x) = (x[@U00C2][@U00B2] + 1)[@U00C2][@U00B7]cos(x[@U00C2][@U00B3]) using both the product and chain rules.
17Find the derivative of f(x) = [@U00E2][@U02C6][@U0161]x[@U00C2][@U00B7]sin(x) using the product rule.
18Which of these is an incorrect application of the product rule for f(x) = u(x)v(x)?
19Given f(x) = (x[@U00C2][@U00B2])(e[@U00CB][@U00A3]), if one mistakenly differentiates it as f'(x) = 2x[@U00C2][@U00B7]e[@U00CB][@U00A3], what step was omitted?
20Differentiate f(x) = x[@U00C2][@U00B7]e[@U00CB][@U00A3][@U00C2][@U00B7]sin(x) by applying the product rule iteratively.
Learning Goals
Study Outcomes
- Understand the concept and definition of the product rule in differentiation.
- Apply the product rule to differentiate functions involving products of two differentiable functions.
- Analyze expressions to correctly identify component functions for the application of the product rule.
- Synthesize different problem-solving approaches when using the product rule in complex scenarios.
- Evaluate the effectiveness of the product rule in solving various calculus problems to build confidence for assessments.
Study Guide
Product Rule Worksheet Cheat Sheet
- Understand the product rule formula - Get cozy with how the derivative of two functions f(x) and g(x) multiplies out to f'(x)g(x) + f(x)g'(x). This is your golden ticket to tackling products in calculus with confidence. Math is Fun: Product Rule
- Practice with different function types - Flex your muscles by applying the product rule to polynomials, trigonometric functions, exponentials, and more to see it in action. Repetition makes perfect, so mix and match functions until it feels like second nature. GeeksforGeeks: Product Rule Formula
- Learn the proof from first principles - Dive into the derivation straight from the definition of a derivative to really own your understanding. Seeing the mechanics helps you connect the dots instead of just memorizing the formula. GeeksforGeeks: Product Rule Derivation
- Extend the rule to multiple functions - Discover how (fgh)' expands to f'gh + fg'h + fgh' and beyond when you have three or more factors. This powerful generalization shows you that calculus can scale to any crease of complexity. Math is Fun: Product Rule
- Spot the difference from the chain rule - Remember: product rule is for multiplying functions, while the chain rule handles nested ones like f(g(x)). Knowing when to use each rule is a must-have skill in your calculus toolkit. GeeksforGeeks: Product vs Chain Rule
- Know its limitations - The product rule only works if each function is differentiable at the point you're examining - no exceptions. Spotting non-differentiable cases early saves you from wild goose chases. GeeksforGeeks: Differentiability Conditions
- Work through practice problems - Reinforce your skills by tackling a variety of exercises, from straightforward to brain-busting. Regular practice helps you anticipate common pitfalls and build unshakeable confidence. GeeksforGeeks: Product Rule Exercises
- Link to the quotient rule - Notice how the quotient rule comes from applying the product rule to f(x) and [g(x)]â»Â¹ - it's all connected! This insight makes both rules feel less like separate hoops and more like a unified strategy. BYJU'S: Product & Quotient Rules
- Use fun mnemonics - Try "first times the derivative of the second plus second times the derivative of the first" to keep it stuck in your brain. A catchy phrase goes a long way when exam day stress hits. Math is Fun: Mnemonic Tips
- Apply it in real-world scenarios - From physics to economics, see how the product rule helps analyze changing quantities in everything from motion to profit. Real examples make abstract formulas feel alive and kickstart your curiosity. BYJU'S: Real-World Applications
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Updated Feb 19, 2026