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Derivatives Practice Tests

These Grade 12 Math practice tests focus on derivatives as instantaneous rates of change, including derivative rules, tangent lines, implicit differentiation, related rates, motion, extrema, concavity, optimization, and inverse functions.

Skills covered

  • Applying power, product, quotient, chain, trigonometric, exponential, and logarithmic derivative rules
  • Interpreting derivatives as slopes, velocity, acceleration, marginal quantities, and monotonicity
  • Using critical numbers, first and second derivative tests, concavity, and the Mean Value Theorem
  • Solving implicit differentiation and related-rates problems with correct units
  • Applying tangent-line approximation, inverse derivatives, parametric derivatives, and accumulation functions

How to use these practice tests

First identify the requested derivative or rate. Then choose the appropriate rule, simplify the expression, and interpret the sign and units in the original context.

For more Grade 12 Math practice, return to the Grade 12 Math category.

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Grade 12 Derivatives Practice Tests

Practice Grade 12 Math questions about derivative definitions, power, product, quotient and chain rules, trigonometric and logarithmic derivatives, implicit differentiation, related rates, extrema, concavity, and tangent lines.

Choose the best answer. Identify the derivative rule or interpretation, calculate carefully, and check signs, units, domains, and critical-point conditions.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

What does f'(a) represent?

Please select an answer first.

Correct answer: A

The correct answer is The instantaneous rate of change at x=a.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: The derivative gives the limiting slope at a single input.
  3. Interpret the result: The correct answer is The instantaneous rate of change at x=a.

The derivative gives the limiting slope at a single input.

2. QuestionMultiple choice

If f'(x)>0 on (1,4), what is f doing there?

Please select an answer first.

Correct answer: B

The correct answer is Increasing.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: A positive derivative indicates increasing output.
  3. Interpret the result: The correct answer is Increasing.

A positive derivative indicates increasing output.

3. QuestionMultiple choice

If f'(x)<0 on (2,6), what is f doing there?

Please select an answer first.

Correct answer: C

The correct answer is Decreasing.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: A negative derivative indicates decreasing output.
  3. Interpret the result: The correct answer is Decreasing.

A negative derivative indicates decreasing output.

4. QuestionMultiple choice

What is a critical number of f?

Please select an answer first.

Correct answer: D

The correct answer is An input in the domain where f'=0 or f' does not exist.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: Critical numbers are possible locations for local extrema or other changes.
  3. Interpret the result: The correct answer is An input in the domain where f'=0 or f' does not exist.

Critical numbers are possible locations for local extrema or other changes.

5. QuestionMultiple choice

What is the slope of the tangent to y=x3y=x^3 at x=2?

Please select an answer first.

Correct answer: A

The correct answer is 12.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: The derivative is 3x^2, so the slope is 3(2^2)=12.
  3. Interpret the result: The correct answer is 12.

The derivative is 3x^2, so the slope is 3(2^2)=12.

6. QuestionMultiple choice

What does a sign change from positive to negative in f' suggest?

Please select an answer first.

Correct answer: B

The correct answer is A local maximum.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: The function changes from increasing to decreasing.
  3. Interpret the result: The correct answer is A local maximum.

The function changes from increasing to decreasing.

7. QuestionMultiple choice

What does a sign change from negative to positive in f' suggest?

Please select an answer first.

Correct answer: C

The correct answer is A local minimum.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: The function changes from decreasing to increasing.
  3. Interpret the result: The correct answer is A local minimum.

The function changes from decreasing to increasing.

8. QuestionMultiple choice

If f''(x)>0 on an interval, the graph is:

Please select an answer first.

Correct answer: D

The correct answer is Concave up.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: A positive second derivative means slopes are increasing.
  3. Interpret the result: The correct answer is Concave up.

A positive second derivative means slopes are increasing.

9. QuestionMultiple choice

What does a change in sign of f'' often indicate?

Please select an answer first.

Correct answer: A

The correct answer is An inflection point.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: A change in concavity occurs when the second derivative changes sign.
  3. Interpret the result: The correct answer is An inflection point.

A change in concavity occurs when the second derivative changes sign.

10. QuestionMultiple choice

What is the tangent line to f at x=a based on f(a) and f'(a)?

Please select an answer first.

Correct answer: B

The correct answer is y−f(a)=f′(a)(x−a)y-f(a)=f'(a)(x-a).

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: Point-slope form uses point (a,f(a)) and slope f'(a).
  3. Interpret the result: The correct answer is y−f(a)=f′(a)(x−a)y-f(a)=f'(a)(x-a).

Point-slope form uses point (a,f(a)) and slope f'(a).

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