Quotient Rule Mastery & Applications

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Quizzes Created: 7682 | Total Attempts: 9,547,133
| Questions: 15 | Updated: Dec 16, 2025
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1) Find the derivative of f(x) = (x² + 3) / (x + 1)

Explanation

Using the Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]². Here u(x) = x² + 3 and u'(x) = 2x. v(x) = x + 1 and v'(x) = 1. Substituting gives [2x(x + 1) - (x² + 3)(1)] / (x + 1)².

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About This Quiz
Quotient Rule Mastery & Applications - Quiz

Ready to take the quotient rule to the next level? In this quiz, you’ll tackle advanced problems that combine the Quotient Rule with the Chain Rule, evaluate derivatives at specific points, and recognize when algebraic simplification can make differentiation easier. You’ll also analyze common student errors and apply the quotient... see morerule in more conceptual situations. By the end, you’ll be solving complex quotient derivatives with confidence and precision.
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2) What is the derivative of g(x) = (5x³ - 2x) / (x² + 4)?

Explanation

For the Quotient Rule, we need u'(x)v(x) - u(x)v'(x) in the numerator. u(x) = 5x³ - 2x gives u'(x) = 15x² - 2. v(x) = x² + 4 gives v'(x) = 2x. The correct setup is [(15x² - 2)(x² + 4) - (5x³ - 2x)(2x)] divided by (x² + 4)².

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3) Find h'(x) for h(x) = sin(x) / cos(x)

Explanation

u(x) = sin(x) so u'(x) = cos(x). v(x) = cos(x) so v'(x) = -sin(x). Applying the Quotient Rule formula [u'(x)v(x) - u(x)v'(x)] / v(x)² gives [cos(x)cos(x) - sin(x)(-sin(x))] / cos²(x).

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4) Differentiate f(x) = eˣ / (x³ + 2x)

Explanation

u(x) = eˣ gives u'(x) = eˣ. v(x) = x³ + 2x gives v'(x) = 3x² + 2. The Quotient Rule requires subtraction in the numerator: [u'(x)v(x) - u(x)v'(x)] / v(x)². Therefore we get [eˣ(x³ + 2x) - eˣ(3x² + 2)] / (x³ + 2x)². A common error is adding instead of subtracting.

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5) What is the derivative of g(x) = √(x) / (x - 5)?

Explanation

Rewrite √x as x^(½). Then u(x) = x^(½) gives u'(x) = (½)x^(-½) = 1/(2√x). v(x) = x - 5 gives v'(x) = 1. Applying the Quotient Rule: [u'(x)v(x) - u(x)v'(x)] / v(x)² = [(1/(2√x))(x - 5) - √x(1)] / (x - 5)².

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6) Find the value of f'(2) for f(x) = (x² + 1) / (x - 3)

Explanation

First find the general derivative using Quotient Rule: u(x) = x² + 1 gives u'(x) = 2x. v(x) = x - 3 gives v'(x) = 1. The derivative is [2x(x - 3) - (x² + 1)(1)] / (x - 3)². Then substitute x = 2: [4(2 - 3) - (4 + 1)(1)] / (2 - 3)² = [4(-1) - 5] / 1 = -9.

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7) Differentiate h(x) = (3x² + 2)⁴ / (x³ - 1)

Explanation

This requires both Chain Rule and Quotient Rule. u(x) = (3x² + 2)⁴. By Chain Rule, u'(x) = 4(3x² + 2)³(6x). v(x) = x³ - 1 gives v'(x) = 3x². Applying Quotient Rule: [u'(x)v(x) - u(x)v'(x)] / v(x)² = [4(3x² + 2)³(6x)(x³ - 1) - (3x² + 2)⁴(3x²)] / (x³ - 1)².

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8) Find the derivative of f(x) = (x² + 5) / √(x + 2)

Explanation

u(x) = x² + 5 gives u'(x) = 2x. v(x) = √(x + 2) = (x + 2)^(½). By Chain Rule, v'(x) = (½)(x + 2)^(-½) = 1/(2√(x + 2)). The Quotient Rule gives [2x√(x + 2) - (x² + 5)(1/(2√(x + 2)))] / [√(x + 2)]². Since [√(x + 2)]² = x + 2, the denominator simplifies to (x + 2).

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9) Which of the following correctly represents the Quotient Rule for f(x) = u(x)/v(x)?

Explanation

The Quotient Rule formula is (low x derivative of high - high x derivative of low) over low squared. In mathematical notation: if f(x) = u(x)/v(x), then f'(x) = [v(x)u'(x) - u(x)v'(x)] / [v(x)]². The order of subtraction matters because the denominator's derivative is subtracted from the numerator's derivative times the denominator.

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10) A student incorrectly derived g(x) = (x³ + 2) / (x² - 1) as g'(x) = [3x²(x² - 1) + (x³ + 2)(2x)] / (x² - 1)². What error was made?

Explanation

The Quotient Rule requires subtraction in the numerator: [u'(x)v(x) - u(x)v'(x)] / [v(x)]². The student's solution shows addition between the two terms in the numerator. The correct derivative should be [3x²(x² - 1) - (x³ + 2)(2x)] / (x² - 1)². This is a common mistake when applying the Quotient Rule.

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11) For which of the following functions would using the Quotient Rule be most appropriate?

Explanation

The Quotient Rule is specifically designed for functions that are ratios of two differentiable functions (one divided by another). Option C is explicitly written as a fraction with numerator 4x + 5 and denominator x² - 3x. Option A would use Product Rule, B would use Chain Rule, and D is a composite function requiring Chain Rule.

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12) When finding the second derivative of f(x) = (2x + 1) / (x - 3), what should be your first step?

Explanation

To find the second derivative f''(x), you must first find the first derivative f'(x). For a quotient function, this means applying the Quotient Rule (or Product Rule with negative exponents) to differentiate f(x) once. Only after obtaining f'(x) can you then differentiate it again to get f''(x). The order of operations matters - you cannot skip directly to the second derivative.

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13) If f(x) = p(x) / q(x) where p(2) = 5, p'(2) = -1, q(2) = 3, and q'(2) = 4, what is f'(2)?

Explanation

The Quotient Rule at a specific point uses the formula [p'(c)q(c) - p(c)q'(c)] / [q(c)]². Substituting c = 2, p(2) = 5, p'(2) = -1, q(2) = 3, q'(2) = 4 gives [(-1)(3) - (5)(4)] / (3)² = (-3 - 20) / 9 = -23/9. This tests understanding of the Quotient Rule's structure without requiring symbolic differentiation.

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14) Which function has a derivative of the form [g'(x)h(x) - g(x)h'(x)] / [h(x)]²?

Explanation

The form [g'(x)h(x) - g(x)h'(x)] / [h(x)]² is the structural template for the Quotient Rule. This pattern specifically identifies a function that is a ratio or quotient of two functions, where g(x) is the numerator function and h(x) is the denominator function. Option B, f(x) = g(x) / h(x), matches this structure exactly. The other options represent product, composition, and sum respectively.

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15) Before applying the Quotient Rule to f(x) = (x³ - 8) / (x - 2), what algebraic step could simplify the process?

Explanation

Recognizing that x³ - 8 is a difference of cubes allows factoring: x³ - 8 = (x - 2)(x² + 2x + 4). The original function becomes f(x) = [(x - 2)(x² + 2x + 4)] / (x - 2). For x ≠ 2, the (x - 2) terms cancel, leaving f(x) = x² + 2x + 4, which is much simpler to differentiate directly using the Power Rule. This demonstrates an important conceptual skill: looking for algebraic simplifications before automatically applying differentiation rules.

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Find the derivative of f(x) = (x² + 3) / (x + 1)
What is the derivative of g(x) = (5x³ - 2x) / (x² + 4)?
Find h'(x) for h(x) = sin(x) / cos(x)
Differentiate f(x) = eˣ / (x³ + 2x)
What is the derivative of g(x) = √(x) / (x - 5)?
Find the value of f'(2) for f(x) = (x² + 1) / (x - 3)
Differentiate h(x) = (3x² + 2)⁴ / (x³ - 1)
Find the derivative of f(x) = (x² + 5) / √(x + 2)
Which of the following correctly represents the Quotient Rule for f(x)...
A student incorrectly derived g(x) = (x³ + 2) / (x² - 1) as g'(x) =...
For which of the following functions would using the Quotient Rule be...
When finding the second derivative of f(x) = (2x + 1) / (x - 3), what...
If f(x) = p(x) / q(x) where p(2) = 5, p'(2) = -1, q(2) = 3, and q'(2)...
Which function has a derivative of the form [g'(x)h(x) - g(x)h'(x)] /...
Before applying the Quotient Rule to f(x) = (x³ - 8) / (x - 2), what...
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