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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

For x2+y2=25x^2+y^2=25, what is dydx\frac{dy}{dx}?

Please select an answer first.

Correct answer: A

The correct answer is −xy-\frac{x}{y}.

  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: Differentiate implicitly: 2x+2y y'=0, so y'=-x/y.
  3. Interpret the result: The correct answer is −xy-\frac{x}{y}.

Differentiate implicitly: 2x+2y y'=0, so y'=-x/y.

2. QuestionMultiple choice

For xy=12xy=12, what is dydx\frac{dy}{dx}?

Please select an answer first.

Correct answer: B

The correct answer is −yx-\frac{y}{x}.

  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: Differentiate x y'+y=0 and solve y'=-y/x.
  3. Interpret the result: The correct answer is −yx-\frac{y}{x}.

Differentiate x y'+y=0 and solve y'=-y/x.

3. QuestionMultiple choice

A circle has radius r with drdt=2\frac{dr}{dt}=2. What is dAdt\frac{dA}{dt} when r=3?

Please select an answer first.

Correct answer: C

The correct answer is 12π.

  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=πr^2, so dA/dt=2πr dr/dt=2π(3)(2)=12π.
  3. Interpret the result: The correct answer is 12π.

A=πr^2, so dA/dt=2πr dr/dt=2π(3)(2)=12π.

4. QuestionMultiple choice

A 13-foot ladder has its base 5 feet from a wall. What is its height?

Please select an answer first.

Correct answer: D

The correct answer is 12 feet.

  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: Use the 5-12-13 right triangle.
  3. Interpret the result: The correct answer is 12 feet.

Use the 5-12-13 right triangle.

5. QuestionMultiple choice

If x2+y2=169x^2+y^2=169, what is the slope at (5,12)?

Please select an answer first.

Correct answer: A

The correct answer is −512-\frac{5}{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 implicit derivative is -x/y, so the slope is -5/12.
  3. Interpret the result: The correct answer is −512-\frac{5}{12}.

The implicit derivative is -x/y, so the slope is -5/12.

6. QuestionMultiple choice

If a sphere has radius r and drdt=1\frac{dr}{dt}=1, what is dVdt\frac{dV}{dt} at r=2?

Please select an answer first.

Correct answer: B

The correct answer is 16π.

  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: V=4πr^3/3, so dV/dt=4πr^2 dr/dt=16π.
  3. Interpret the result: The correct answer is 16π.

V=4πr^3/3, so dV/dt=4πr^2 dr/dt=16π.

7. QuestionMultiple choice

What does a related-rates equation connect?

Please select an answer first.

Correct answer: C

The correct answer is Rates of changing quantities linked by a common relationship.

  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: Different variables are related through an equation before differentiating with respect to time.
  3. Interpret the result: The correct answer is Rates of changing quantities linked by a common relationship.

Different variables are related through an equation before differentiating with respect to time.

8. QuestionMultiple choice

If s(t)=t2+1s(t)=t^2+1, what is dsdt\frac{ds}{dt} at t=4?

Please select an answer first.

Correct answer: D

The correct answer is 8.

  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: Differentiate s(t): ds/dt=2t, then evaluate at 4.
  3. Interpret the result: The correct answer is 8.

Differentiate s(t): ds/dt=2t, then evaluate at 4.

9. QuestionMultiple choice

If revenue is R(q)=100q−q2R(q)=100q-q^2, what is marginal revenue at q=30?

Please select an answer first.

Correct answer: A

The correct answer is 40.

  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: R'(q)=100-2q, so R'(30)=40.
  3. Interpret the result: The correct answer is 40.

R'(q)=100-2q, so R'(30)=40.

10. QuestionMultiple choice

What unit does a related rate dVdt\frac{dV}{dt} have if V is in cubic meters and t is in seconds?

Please select an answer first.

Correct answer: B

The correct answer is Cubic meters per second.

  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 rate unit follows the changing quantity divided by time.
  3. Interpret the result: The correct answer is Cubic meters per second.

The rate unit follows the changing quantity divided by time.

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