NumicornNumicornLearn, solve, progress

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.

Select a test

5 Tests
1. Test2. Test3. Test4. Test5. Test

4. Test

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 is the absolute maximum of f(x)=x2f(x)=x^2 on [-2,1]?

Please select an answer first.

Correct answer: A

The correct answer is 4.

  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: Check endpoints and the critical point: f(-2)=4, f(1)=1, and f(0)=0.
  3. Interpret the result: The correct answer is 4.

Check endpoints and the critical point: f(-2)=4, f(1)=1, and f(0)=0.

2. QuestionMultiple choice

What theorem guarantees an absolute maximum and minimum for a continuous function on a closed interval?

Please select an answer first.

Correct answer: B

The correct answer is The Extreme Value Theorem.

  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: Continuity on a closed and bounded interval guarantees both absolute extrema.
  3. Interpret the result: The correct answer is The Extreme Value Theorem.

Continuity on a closed and bounded interval guarantees both absolute extrema.

3. QuestionMultiple choice

What is the second derivative test used to classify?

Please select an answer first.

Correct answer: C

The correct answer is A critical point as a local maximum or 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: At a critical point, the sign of f'' can indicate local concavity and type.
  3. Interpret the result: The correct answer is A critical point as a local maximum or minimum.

At a critical point, the sign of f'' can indicate local concavity and type.

4. QuestionMultiple choice

If f'(c)=0 and f''(c)<0, what is likely at c?

Please select an answer first.

Correct answer: D

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: Negative second derivative means the graph is concave down at the stationary point.
  3. Interpret the result: The correct answer is A local maximum.

Negative second derivative means the graph is concave down at the stationary point.

5. QuestionMultiple choice

What does Rolle's Theorem guarantee when its conditions hold?

Please select an answer first.

Correct answer: A

The correct answer is At least one point has f'(c)=0.

  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: Equal endpoint values force a horizontal tangent somewhere inside the interval.
  3. Interpret the result: The correct answer is At least one point has f'(c)=0.

Equal endpoint values force a horizontal tangent somewhere inside the interval.

6. QuestionMultiple choice

What does the Mean Value Theorem say about a differentiable function?

Please select an answer first.

Correct answer: B

The correct answer is Some tangent slope equals the secant slope.

  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: There is c in the interval with f'(c)=(f(b)-f(a))/(b-a).
  3. Interpret the result: The correct answer is Some tangent slope equals the secant slope.

There is c in the interval with f'(c)=(f(b)-f(a))/(b-a).

7. QuestionMultiple choice

What dimensions maximize the area of a rectangle with perimeter 24?

Please select an answer first.

Correct answer: C

The correct answer is 6 by 6.

  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 square maximizes area for a fixed perimeter; each side is 24/4=6.
  3. Interpret the result: The correct answer is 6 by 6.

A square maximizes area for a fixed perimeter; each side is 24/4=6.

8. QuestionMultiple choice

If cost C(q) is increasing and C'(q)>0, what does C'(q) measure?

Please select an answer first.

Correct answer: D

The correct answer is Marginal cost.

  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 approximate cost of one additional unit.
  3. Interpret the result: The correct answer is Marginal cost.

The derivative gives the approximate cost of one additional unit.

9. QuestionMultiple choice

What is the linearization of f at x=a?

Please select an answer first.

Correct answer: A

The correct answer is L(x)=f(a)+f′(a)(x−a)L(x)=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: Linearization uses the tangent line at x=a.
  3. Interpret the result: The correct answer is L(x)=f(a)+f′(a)(x−a)L(x)=f(a)+f'(a)(x-a).

Linearization uses the tangent line at x=a.

10. QuestionMultiple choice

Why is a derivative useful in optimization?

Please select an answer first.

Correct answer: B

The correct answer is It locates candidate extrema where the rate of change is zero or undefined.

  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 and endpoints provide candidates for maximum or minimum values.
  3. Interpret the result: The correct answer is It locates candidate extrema where the rate of change is zero or undefined.

Critical numbers and endpoints provide candidates for maximum or minimum values.

Previous testNext test