Product Rule: Mixed Concepts & Common Pitfalls

Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7682 | Total Attempts: 9,547,133
| Questions: 15 | Updated: Dec 16, 2025
Please wait...
Question 1 / 15
0 %
0/100
Score 0/100
1) Find the derivative of f(x) = (x⁴ + 2x²)(3x³ + 5x²)

Explanation

Using the Product Rule: u(x) = x⁴ + 2x², u'(x) = 4x³ + 4x; v(x) = 3x³ + 5x², v'(x) = 9x² + 10x
f'(x) = u'(x)v(x) + u(x)v'(x) = (4x³ + 4x)(3x³ + 5x²) + (x⁴ + 2x²)(9x² + 10x)
Submit
Please wait...
About This Quiz
Product Rule: Mixed Concepts & Common Pitfalls - Quiz

Ready for a challenge? In this quiz, you’ll tackle advanced Product Rule problems involving large polynomials and extended products. You’ll also apply the rule to real-world scenarios and conceptual questions that test your understanding—not just computation. This quiz helps you see how algebra and calculus work together in deeper ways.

2)
You may optionally provide this to label your report, leaderboard, or certificate.
2) If g(x) = (5x² - 3x + 1)(2x⁴ + x³), what is g'(x)?

Explanation

Using the Product Rule: u(x) = 5x² - 3x + 1, u'(x) = 10x - 3; v(x) = 2x⁴ + x³, v'(x) = 8x³ + 3x²
g'(x) = u'(x)v(x) + u(x)v'(x) = (10x - 3)(2x⁴ + x³) + (5x² - 3x + 1)(8x³ + 3x²)
Submit
3) What is the derivative of h(x) = (x³ - 4x)(x⁵ + 3x³ - x)?

Explanation

Using the Product Rule: u(x) = x³ - 4x, u'(x) = 3x² - 4; v(x) = x⁵ + 3x³ - x, v'(x) = 5x⁴ + 9x² - 1
h'(x) = u'(x)v(x) + u(x)v'(x) = (3x² - 4)(x⁵ + 3x³ - x) + (x³ - 4x)(5x⁴ + 9x² - 1)
Submit
4) Calculate f'(x) for f(x) = (7x³ + 2x² - 5x)(4x² + 3x + 2)

Explanation

Using the Product Rule: u(x) = 7x³ + 2x² - 5x, u'(x) = 21x² + 4x - 5; v(x) = 4x² + 3x + 2, v'(x) = 8x + 3
f'(x) = u'(x)v(x) + u(x)v'(x) = (21x² + 4x - 5)(4x² + 3x + 2) + (7x³ + 2x² - 5x)(8x + 3)
Submit
5) If y = (x⁴ - x² + 3)(x³ + 2x), what is dy/dx?

Explanation

Using the Product Rule: u(x) = x⁴ - x² + 3, u'(x) = 4x³ - 2x; v(x) = x³ + 2x, v'(x) = 3x² + 2
dy/dx = u'(x)v(x) + u(x)v'(x) = (4x³ - 2x)(x³ + 2x) + (x⁴ - x² + 3)(3x² + 2)
Submit
6) Find the derivative of g(x) = (x⁵ + 4x³ - x²)(3x⁴ + 2x² + 1)

Explanation

Using the Product Rule: u(x) = x⁵ + 4x³ - x², u'(x) = 5x⁴ + 12x³ - 2x; v(x) = 3x⁴ + 2x² + 1, v'(x) = 12x³ + 4x
g'(x) = u'(x)v(x) + u(x)v'(x) = (5x⁴ + 12x³ - 2x)(3x⁴ + 2x² + 1) + (x⁵ + 4x³ - x²)(12x³ + 4x)
Submit
7) What is h'(x) for h(x) = (2x⁶ - 5x⁴ + 3x²)(x⁵ + 4x³ + 2x)?

Explanation

Using the Product Rule: u(x) = 2x⁶ - 5x⁴ + 3x², u'(x) = 12x⁵ - 20x³ + 6x; v(x) = x⁵ + 4x³ + 2x, v'(x) = 5x⁴ + 12x² + 2
h'(x) = u'(x)v(x) + u(x)v'(x) = (12x⁵ - 20x³ + 6x)(x⁵ + 4x³ + 2x) + (2x⁶ - 5x⁴ + 3x²)(5x⁴ + 12x² + 2)
Submit
8) Find f'(x) where f(x) = (3x⁴ - 2x³ + x)(6x³ + 4x - 1)

Explanation

Using the Product Rule: u(x) = 3x⁴ - 2x³ + x, u'(x) = 12x³ - 6x² + 1; v(x) = 6x³ + 4x - 1, v'(x) = 18x² + 4
f'(x) = u'(x)v(x) + u(x)v'(x) = (12x³ - 6x² + 1)(6x³ + 4x - 1) + (3x⁴ - 2x³ + x)(18x² + 4)
Submit
9) Calculate g'(x) for g(x) = (x⁷ - 3x⁵ + 2x³)(2x⁴ + 3x² + 5)

Explanation

Using the Product Rule: u(x) = x⁷ - 3x⁵ + 2x³, u'(x) = 7x⁶ - 15x⁴ + 6x²; v(x) = 2x⁴ + 3x² + 5, v'(x) = 8x³ + 6x
g'(x) = u'(x)v(x) + u(x)v'(x) = (7x⁶ - 15x⁴ + 6x²)(2x⁴ + 3x² + 5) + (x⁷ - 3x⁵ + 2x³)(8x³ + 6x)
Submit
10) What is the product rule formula for finding the derivative of f(x) = u(x)v(x)?

Explanation

The product rule states that when differentiating a product of two functions, you take the derivative of the first function times the second function, plus the first function times the derivative of the second function. This can be remembered as "first times derivative of second, plus second times derivative of first.

Submit
11) The Product Rule is derived from the limit definition of the derivative. What is the correct starting point for this derivation?

Explanation

The derivation of the Product Rule begins with the limit definition of the derivative applied to the product function f(x)=u(x)v(x). This gives us option A, which can then be manipulated algebraically (by adding and subtracting u(x+h)v(x)) to eventually arrive at the Product Rule formula f'(x)=u'(x)v(x)+u(x)v'(x).

Submit
12) A company's cost function is C(x) = (a + bx²)(c + dx). What does C'(x) represent?

Explanation

C'(x) represents the derivative of the total cost function with respect to quantity x. In economics, this is called the marginal cost, which represents the additional cost of producing one more unit. It tells us how quickly the total cost is changing as production levels change.

Submit
13) When differentiating f(x) = x³eˣ, which of these is NOT a correct step?

Explanation

The Product Rule is f'(x) = u'(x)v(x) + u(x)v'(x), not minus. Option C shows a subtraction, which would be incorrect. This is a common sign error - students sometimes confuse the Product Rule with the Quotient Rule, which has a subtraction in the numerator. 

Submit
14) If f(x) = u(x)v(x) and both u and v are differentiable, which statement about f'(x) is FALSE?

Explanation

The Product Rule states that f'(x) = u'(x)v(x) + u(x)v'(x), NOT u'(x)v'(x). This is a common misconception. Option C is false because the derivative of a product is not simply the product of the derivatives. For example, if u(x) = x and v(x) = x, then f(x) = x², f'(x) = 2x, but u'(x)v'(x) = 1 × 1 = 1, which is incorrect.

Submit
15) Consider p(x) = f(x)g(x)h(x). A student writes p'(x) = f'(x)g'(x)h(x) + f(x)g'(x)h'(x). What error did they make?

Explanation

The correct extended Product Rule for three functions is p'(x) = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x). The student made two errors:

1. They forgot the middle term f(x)g'(x)h(x)

2. They incorrectly multiplied derivatives in the first term (f'(x)g'(x)h(x)) instead of multiplying each derivative with the original functions of the others.

Submit
×
Saved
Thank you for your feedback!
View My Results
Cancel
  • All
    All (15)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
Find the derivative of f(x) = (x⁴ + 2x²)(3x³ + 5x²)
If g(x) = (5x² - 3x + 1)(2x⁴ + x³), what is g'(x)?
What is the derivative of h(x) = (x³ - 4x)(x⁵ + 3x³ - x)?
Calculate f'(x) for f(x) = (7x³ + 2x² - 5x)(4x² + 3x + 2)
If y = (x⁴ - x² + 3)(x³ + 2x), what is dy/dx?
Find the derivative of g(x) = (x⁵ + 4x³ - x²)(3x⁴ + 2x² + 1)
What is h'(x) for h(x) = (2x⁶ - 5x⁴ + 3x²)(x⁵ + 4x³ + 2x)?
Find f'(x) where f(x) = (3x⁴ - 2x³ + x)(6x³ + 4x - 1)
Calculate g'(x) for g(x) = (x⁷ - 3x⁵ + 2x³)(2x⁴ + 3x² + 5)
What is the product rule formula for finding the derivative of f(x) =...
The Product Rule is derived from the limit definition of the...
A company's cost function is C(x) = (a + bx²)(c + dx). What...
When differentiating f(x) = x³eˣ, which of these is NOT a...
If f(x) = u(x)v(x) and both u and v are differentiable, which...
Consider p(x) = f(x)g(x)h(x). A student writes p'(x) =...
Alert!

Advertisement