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Unit 2 of 5

Unit 2: Derivatives

Study guide for CLEP CLEP Calculus — Unit 2: Derivatives. Practice questions, key concepts, and exam tips.

76

Practice Questions

17

Flashcards

11

Key Topics

Key Concepts to Study

derivative definition
power/product/quotient/chain rules
implicit differentiation
trig/exponential/log derivatives
Mean Value Theorem
L'Hopital's Rule
first derivative test for extrema
second derivative test
absolute vs relative extrema
concavity and inflection points
curve sketching using derivatives

Sample Practice Questions

Try these 5 questions from this unit. Sign up for full access to all 76.

Q1EASY

If f(x) = 3x23x^{2}, what is the derivative of f(x) with respect to x?

A.

x2x^{2}

B.

3x23x^{2}

C.

9x

D.

6x

E.

6

Show Answer

Answer: D — The derivative of f(x) with respect to x is found using the power rule, which states that if f(x) = $x^{n}$, then f'(x) = $nx^{n-1}$. Applying this rule to f(x) = $3x^{2}$ yields f'(x) = $3 \cdot 2x^{2-1}$ = $6x$.

Q2MEDIUM

If f(x) = 3x23x^{2}, what is the average rate of change of f(x) over the interval [1, 3]?

A.

0

B.

3

C.

6

D.

12

E.

15

Show Answer

Answer: D — The average rate of change is calculated as (f(3) - f(1)) / (3 - 1) = (27 - 3) / 2 = 12.

Q3HARD

A state's education budget is modeled by the function C(x) = 2x22x^{2} + 5x + 1, where C(x) is the number of graduates and x is the public investment in education in millions of dollars. What is the instantaneous rate of change of graduates with respect to the investment when x = 4?

A.

2(4)2(4)^{2} + 5(4)

B.

18

C.

2(4) + 5

D.

21

E.

19

Show Answer

Answer: D — The derivative of C(x) is C'(x) = 4x + 5. Evaluating at x = 4, C'(4) = 4(4) + 5 = 21.

Q4HARD

If f(x) = 2x+1x2−4\frac{2x + 1}{x^2 - 4}, find f'(x) using the quotient rule.

A.

f'(x) = (2 - (2x + 1)(2)) / (x2x^{2} - 4)^2

B.

f'(x) = (2 - (2x + 1)(2x)) / (x2x^{2} - 4)^2

C.

f'(x) = (2(x2x^{2} - 4) + (2x + 1)(2x)) / (x2x^{2} - 4)^2

D.

f'(x) = (2(x2x^{2} - 4) - (2x + 1)(2x)) / (x2x^{2} - 4)^2

E.

f'(x) = (2(x2x^{2} - 4) + (2x + 1)(2x)) / (x2x^{2} - 4)

Show Answer

Answer: D — The quotient rule states that if f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x)) / (h(x))^2, so f'(x) = (2(x^2 - 4) - (2x + 1)(2x)) / (x^2 - 4)^2 is correct because it applies this rule with g(x) = 2x + 1, h(x) = x^2 - 4, g'(x) = 2, and h'(x) = 2x.

Q5HARD

If f(x) = 3x23x^{2}, what is the instantaneous rate of change of f at x = 2?

A.

lim (h → 0) [f(2 + h) - f(2)]/h = 12

B.

lim (h → 0) [f(2 + h) - f(2)]/h = 6

C.

lim (h → 0) [f(2 + h) - f(2)] = 12

D.

lim (h → 0) [f(2 + h) - f(2)]/h = 0

E.

lim (h → 0) [f(2 + h) - f(2)]/h = -12

Show Answer

Answer: A — To find the instantaneous rate of change, we use the limit definition of a derivative. For f(x) = $3x^{2}$, f'(x) = lim (h → 0) [f(x + h) - f(x)]/h. At x = 2, f'(2) = lim (h → 0) [3(2 + h)^2 - 3(2)^2]/h = lim (h → 0) [3(4 + 4h + $h^{2}$) - 12]/h = lim (h → 0) [12 + 12h + $3h^{2}$ - 12]/h = lim (h → 0) [12h + $3h^{2}$]/h = lim (h → 0) [12 + 3h] = 12.

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Study Tips for Unit 2: Derivatives

  • Focus on understanding concepts, not memorizing facts — CLEP tests application
  • Practice with timed questions to build exam-day speed
  • Review explanations for wrong answers — they reveal common misconceptions
  • Use flashcards for key terms, practice questions for deeper understanding

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