Product 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) The product rule states that if f(x)=g(x)·h(x), then f'(x) equals:

Explanation

The product rule is one of the fundamental differentiation rules. It states that the derivative of a product is the derivative of the first function times the second function, plus the first function times the derivative of the second function. This is often remembered as "d(first)·second + first·d(second)."

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About This Quiz
Product Rule Mastery & Applications - Quiz

Ready to think like a calculus expert? This quiz pushes your understanding of the Product Rule through advanced applications, including function ratios, implicit reasoning, and real-world interpretations like marginal change. You’ll analyze derivatives conceptually, work with given function values, and apply the rule to multi-function products. It’s the perfect wrap-up... see moreto mastering the Product Rule with confidence.
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2) If f(x)=x·g(x), what is f'(x)/f(x)?

Explanation

First, find f'(x) using the Product Rule: f'(x) = 1·g(x) + x·g'(x). Now divide by f(x) = x·g(x):
[g(x) + x·g'(x)] / [x·g(x)] = g(x)/[x·g(x)] + [x·g'(x)]/[x·g(x)] = 1/x + g'(x)/g(x).
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3) What is the derivative of y=eˣ·x²?

Explanation

Using the product rule with u = eˣ and v = x²:
u' = eˣ
v' = 2x
y' = u'·v + u·v' = eˣ·x² + eˣ·2x
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4) Find the derivative of g(x)=(5x+2)(3x-1).

Explanation

Using the product rule with u = 5x + 2 and v = 3x - 1:
u' = 5
v' = 3
g'(x) = u'·v + u·v' = 5(3x - 1) + (5x + 2)·3 This simplifies to 15x - 5 + 15x + 6 = 30x + 1, but the product rule application is shown in option a.
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5) What is the derivative of h(x)=x³·sin(x)?

Explanation

Using the product rule with u = x³ and v = sin(x):
u' = 3x²
v' = cos(x)
h'(x) = u'·v + u·v' = 3x²·sin(x) + x³·cos(x)
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6) If f(x)=x·ln(x), what is f'(e)?

Explanation

First find f'(x) using the product rule:
u = x, u' = 1
v = ln(x), v' = 1/x
f'(x) = 1·ln(x) + x·(1/x) = ln(x) + 1
Now evaluate at x = e:
f'(e) = ln(e) + 1 = 1 + 1 = 2
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7) Find the derivative of f(x)=cos(x)·eˣ.

Explanation

Using the product rule with u = cos(x) and v = eˣ:
u' = -sin(x)
v' = eˣ
f'(x) = u'·v + u·v' = -sin(x)·eˣ + cos(x)·eˣ
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8) When would the product rule give f'(x)=0?

Explanation

The product rule gives f'(x) = u'(x)·v(x) + u(x)·v'(x). For this to equal zero, we need u'(x)·v(x) + u(x)·v'(x) = 0, which means u'(x)·v(x) = -u(x)·v'(x). Option d would also work if both derivatives are zero, but this is a special case. Option c is incorrect because if both functions are constants, we wouldn't use the product rule (the derivative of a constant product is just 0). Option b is the most general condition.
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9) What is the derivative of p(x)=(x+1)(x²+x)?

Explanation

Using the product rule with u = x + 1 and v = x² + x:
u' = 1
v' = 2x + 1
p'(x) = u'·v + u·v' = 1(x² + x) + (x + 1)(2x + 1)
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10) Find the derivative of q(x)=x⁵·cos(x).

Explanation

Using the product rule with u = x⁵ and v = cos(x):
u' = 5x⁴
v' = -sin(x)
q'(x) = u'·v + u·v' = 5x⁴·cos(x) + x⁵·(-sin(x)) = 5x⁴·cos(x) - x⁵·sin(x)
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11) Let h(x)=x²f(x). Given f(2)=3 and f'(2)=-4, compute h'(2).

Explanation

We treat x² as the first function u(x) and f(x) as the second function v(x). u(x) = x², so u'(x) = 2x. v(x) = f(x), so v'(x) = f'(x). The Product Rule gives h'(x) = 2xf(x) + x²f'(x). At x = 2: h'(2) = 2(2)f(2) + (2)²f'(2) = 4(3) + 4(-4) = 12 - 16 = -4.
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12) The graphs of f(x) and g(x) intersect at x=2 with f(2)=g(2)=3, f'(2)=1, g'(2)=-2. Find slope of h(x)=f(x)g(x) at x=2.

Explanation

We are given f(2) = 3, g(2) = 3, f'(2) = 1, and g'(2) = -2. We need to find h'(2). Using the Product Rule: h'(2) = f'(2)g(2) + f(2)g'(2) = (1)(3) + (3)(-2) = 3 - 6 = -3.
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13) Find the derivative of r(x)=ln(x)·sin(x).

Explanation

Using the product rule with u = ln(x) and v = sin(x):
u' = 1/x
v' = cos(x)
r'(x) = u'·v + u·v' = (1/x)·sin(x) + ln(x)·cos(x)
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14) Differentiate y = x f(x) g(x).

Explanation

We can group terms to use the Product Rule. Let u = x and v = f(x)g(x). Then u' = 1. To find v', we use the Product Rule on f(x)g(x), so v' = f'(x)g(x) + f(x)g'(x).
Now apply the Product Rule to the whole expression: y' = u'v + uv' = 1·[f(x)g(x)] + x·[f'(x)g(x) + f(x)g'(x)]. Distributing the x gives f(x)g(x) + xf'(x)g(x) + xf(x)g'(x).
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15) Which statement about the product rule is FALSE?

Explanation

This statement is false because the product rule applies to ANY two differentiable functions, not just polynomials. We can use it with exponential functions, trigonometric functions, logarithmic functions, or any combination. Options a, c, and d are all true statements about the product rule.
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The product rule states that if f(x)=g(x)·h(x), then f'(x) equals:
If f(x)=x·g(x), what is f'(x)/f(x)?
What is the derivative of y=eˣ·x²?
Find the derivative of g(x)=(5x+2)(3x-1).
What is the derivative of h(x)=x³·sin(x)?
If f(x)=x·ln(x), what is f'(e)?
Find the derivative of f(x)=cos(x)·eˣ.
When would the product rule give f'(x)=0?
What is the derivative of p(x)=(x+1)(x²+x)?
Find the derivative of q(x)=x⁵·cos(x).
Let h(x)=x²f(x). Given f(2)=3 and f'(2)=-4, compute h'(2).
The graphs of f(x) and g(x) intersect at x=2 with f(2)=g(2)=3,...
Find the derivative of r(x)=ln(x)·sin(x).
Differentiate y = x f(x) g(x).
Which statement about the product rule is FALSE?
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