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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.

89

Practice Questions

9

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 89.

Q1MEDIUM

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

A) 6x
B) $3x^{2}$
C) 9x
D) $x^{2}$
E) 6
Show Answer

Answer: A — The derivative of f(x) = $3x^{2}$ is f'(x) = 6x, using the power rule of differentiation which states that if f(x) = x^n, then f'(x) = nx^(n-1). In this case, n = 2, so f'(x) = 3*2*x^(2-1) = 6x.

Q2MEDIUM

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

A) 12
B) 6
C) 3
D) 0
E) 15
Show Answer

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

Q3MEDIUM

If the difference quotient for a function f(x) is given by (f(4) - f(3)) / (4 - 3) = 12.8 - 11.2, what can be inferred about the rate of change of the function at x = 3?

A) The rate of change is increasing at x = 3.
B) The rate of change is 1.6 at x = 3.
C) The rate of change is decreasing at x = 3.
D) The rate of change cannot be determined with the given information.
E) The rate of change is -1.6 at x = 3
Show Answer

Answer: B — The difference quotient gives the average rate of change between x = 3 and x = 4, which is (f(4) - f(3)) / (4 - 3) = 1.6.

Q4MEDIUM

A state's education budget is modeled by the function C(x) = $2x^{2}$ + 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) 21
B) 18
C) 2(4) + 5
D) 2(4)^2 + 5(4)
E) 19
Show Answer

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

Q5HARD

What is the derivative of y = arcsin(x)?

A) 1/$\sqrt{1-$x^{2}$}$
B) 1/$\sqrt{1+$x^{2}$}$
C) x/$\sqrt{1-$x^{2}$}$
D) x/$\sqrt{1+$x^{2}$}$
E) 1/(1-$x^{2}$)
Show Answer

Answer: A — "1/$\sqrt{1-$x^{2}$}$" is correct because it's the derivative of the inverse sine function.

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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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