Product Rule with Polynomial Products

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Alva Benedict B. is an experienced mathematician and math content developer with over 15 years of teaching and tutoring experience across high school, undergraduate, and test prep levels. He specializes in Algebra, Calculus, and Statistics, and holds advanced academic training in Mathematics with extensive expertise in LaTeX-based math content development.
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| Questions: 15 | Updated: Jan 29, 2026
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1) Find the derivative of f(x) = (2x³ - 5x)(4x² + 3x)

Explanation

Using the Product Rule: u(x) = 2x³ - 5x, u'(x) = 6x² - 5; v(x) = 4x² + 3x, v'(x) = 8x + 3
f'(x) = u'(x)v(x) + u(x)v'(x) = (6x² - 5)(4x² + 3x) + (2x³ - 5x)(8x + 3)
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About This Quiz
Product Rule With Polynomial Products - Quiz

Think you’ve mastered the basics? This quiz builds on your Product Rule skills with more complex polynomial expressions and longer algebraic steps. You’ll work through higher-degree functions, carefully track each derivative, and strengthen your accuracy. It’s a great way to sharpen your technique and reduce common sign and setup errors.

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2) If g(x) = (x² + 6)(3x⁴ - 2x²), what is g'(x)?

Explanation

Using the Product Rule: u(x) = x² + 6, u'(x) = 2x; v(x) = 3x⁴ - 2x², v'(x) = 12x³ - 4x
g'(x) = u'(x)v(x) + u(x)v'(x) = 2x(3x⁴ - 2x²) + (x² + 6)(12x³ - 4x)
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3) What is the derivative of h(x) = (5x + 2)(x³ - 7x)?

Explanation

Using the Product Rule: u(x) = 5x + 2, u'(x) = 5; v(x) = x³ - 7x, v'(x) = 3x² - 7
h'(x) = u'(x)v(x) + u(x)v'(x) = 5(x³ - 7x) + (5x + 2)(3x² - 7)
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4) Find the derivative of f(x) = (x⁴ + 3x²)(2x³ + 4x² - 1)

Explanation

Using the Product Rule: u(x) = x⁴ + 3x², u'(x) = 4x³ + 6x; v(x) = 2x³ + 4x² - 1, v'(x) = 6x² + 8x
f'(x) = u'(x)v(x) + u(x)v'(x) = (4x³ + 6x)(2x³ + 4x² - 1) + (x⁴ + 3x²)(6x² + 8x)
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5) If y = (3x⁵ - x²)(6x + 1), what is dy/dx?

Explanation

Using the Product Rule: u(x) = 3x⁵ - x², u'(x) = 15x⁴ - 2x; v(x) = 6x + 1, v'(x) = 6
dy/dx = u'(x)v(x) + u(x)v'(x) = (15x⁴ - 2x)(6x + 1) + (3x⁵ - x²)(6)
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6) Calculate the derivative of f(x) = (x² - 3x + 2)(x⁴ + 2x³)

Explanation

Using the Product Rule: u(x) = x² - 3x + 2, u'(x) = 2x - 3; v(x) = x⁴ + 2x³, v'(x) = 4x³ + 6x²
f'(x) = u'(x)v(x) + u(x)v'(x) = (2x - 3)(x⁴ + 2x³) + (x² - 3x + 2)(4x³ + 6x²)
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7) Find h'(x) for h(x) = (4x⁶ - 2x⁴ + x)(3x² - 5)

Explanation

Using the Product Rule: u(x) = 4x⁶ - 2x⁴ + x, u'(x) = 24x⁵ - 8x³ + 1; v(x) = 3x² - 5, v'(x) = 6x
h'(x) = u'(x)v(x) + u(x)v'(x) = (24x⁵ - 8x³ + 1)(3x² - 5) + (4x⁶ - 2x⁴ + x)(6x)
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8) What is the derivative of g(x) = (7x - 4)(x⁵ + 3x³ - 2x)?

Explanation

Using the Product Rule: u(x) = 7x - 4, u'(x) = 7; v(x) = x⁵ + 3x³ - 2x, v'(x) = 5x⁴ + 9x² - 2
g'(x) = u'(x)v(x) + u(x)v'(x) = 7(x⁵ + 3x³ - 2x) + (7x - 4)(5x⁴ + 9x² - 2)
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9) Find the derivative of f(x) = (x³ + 2x² - x)(4x² + 5x + 1)

Explanation

Using the Product Rule: u(x) = x³ + 2x² - x, u'(x) = 3x² + 4x - 1; v(x) = 4x² + 5x + 1, v'(x) = 8x + 5
f'(x) = u'(x)v(x) + u(x)v'(x) = (3x² + 4x - 1)(4x² + 5x + 1) + (x³ + 2x² - x)(8x + 5)
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10) If h(x) = (2x⁴ - 6x³ + 5)(x⁵ - 3x²), what is h'(x)?

Explanation

Using the Product Rule: u(x) = 2x⁴ - 6x³ + 5, u'(x) = 8x³ - 18x²; v(x) = x⁵ - 3x², v'(x) = 5x³ - 6x
h'(x) = u'(x)v(x) + u(x)v'(x) = (8x³ - 18x²)(x⁵ - 3x²) + (2x⁴ - 6x³ + 5)(5x³ - 6x)
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11) Some students think they can differentiate (x + 1)(x + 2) by expanding. Why might they be correct?

Explanation

For polynomials, expanding first can be valid and often simpler. After expansion: f(x) = (x + 1)(x + 2) = x² + 3x + 2, then f'(x) = 2x + 3. Using Product Rule directly: u(x) = x + 1, v(x) = x + 2, u'(x) = 1, v'(x) = 1, f'(x) = 1(x + 2) + (x + 1)1 = x + 2 + x + 1 = 2x + 3. Both methods yield the same result. For simple, low-degree polynomials like this one, expanding first can be more efficient. However, for polynomials with higher degrees, the Product Rule is almost always the more efficient and less error-prone method.
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12) A particle's position is s(t) = (t² + 1)(t³ - 2t). What is velocity at t=1?

Explanation

Velocity is the derivative of position: v(t) = s'(t)
Using Product Rule: u(t) = t² + 1, u'(t) = 2t; v(t) = t³ - 2t, v'(t) = 3t² - 2
v(t) = 2t(t³ - 2t) + (t² + 1)(3t² - 2) = 2t⁴ - 4t² + 3t⁴ - 2t² + 3t² - 2 = 5t⁴ - 3t² - 2
v(1) = 5(1) - 3(1) - 2 = 5 - 3 - 2 = 0 m/s
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13) For which scenario is Product Rule NOT most efficient?

Explanation

Function C represents a composite function (a function of a function). For f(x) = (x² + 1)4, the Chain Rule is more efficient than expanding or using the Product Rule.  The other functions explicitly involve products of different functions where the Product Rule is appropriate.
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14) Which function requires the Product Rule?

Explanation

Function A: (x² + 1)³ requires the Chain Rule (power of a function).
Function B: x² + 1 is a sum, not a product, so use Power Rule and Sum Rule.
Function C: x³ - 5x is also a sum, not a product.
Function D: x²sin(x) is a product of x² and sin(x), so it requires the Product Rule. This is the only option that is explicitly a product of two different functions.
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15) The Product Rule can be extended to four functions. What is d/dx[uvwz]?

Explanation

The extended Product Rule for n functions states that d/dx[u₁u₂...uₙ] = u₁'u₂...uₙ + u₁u₂'u₃...uₙ + ... + u₁u₂...uₙ'ₙ. For four functions: d/dx[uvwz] = u'vwz + uv'wz + uvw'z + uvwz', where each term differentiates exactly one of the four functions while keeping the others constant. 

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Alva Benedict B. |PhD
College Expert
Alva Benedict B. is an experienced mathematician and math content developer with over 15 years of teaching and tutoring experience across high school, undergraduate, and test prep levels. He specializes in Algebra, Calculus, and Statistics, and holds advanced academic training in Mathematics with extensive expertise in LaTeX-based math content development.
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Find the derivative of f(x) = (2x³ - 5x)(4x² + 3x)
If g(x) = (x² + 6)(3x⁴ - 2x²), what is g'(x)?
What is the derivative of h(x) = (5x + 2)(x³ - 7x)?
Find the derivative of f(x) = (x⁴ + 3x²)(2x³ + 4x² - 1)
If y = (3x⁵ - x²)(6x + 1), what is dy/dx?
Calculate the derivative of f(x) = (x² - 3x + 2)(x⁴ + 2x³)
Find h'(x) for h(x) = (4x⁶ - 2x⁴ + x)(3x² - 5)
What is the derivative of g(x) = (7x - 4)(x⁵ + 3x³ - 2x)?
Find the derivative of f(x) = (x³ + 2x² - x)(4x² + 5x + 1)
If h(x) = (2x⁴ - 6x³ + 5)(x⁵ - 3x²), what is h'(x)?
Some students think they can differentiate (x + 1)(x + 2) by...
A particle's position is s(t) = (t² + 1)(t³ - 2t). What is velocity...
For which scenario is Product Rule NOT most efficient?
Which function requires the Product Rule?
The Product Rule can be extended to four functions. What is...
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