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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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1. 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 ddx(x6)\frac{d}{dx}(x^6)?

Please select an answer first.

Correct answer: A

The correct answer is 6x56x^5.

  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: Apply the power rule and reduce the exponent by one.
  3. Interpret the result: The correct answer is 6x56x^5.

Apply the power rule and reduce the exponent by one.

2. QuestionMultiple choice

What is ddx(3x4−5x2+7)\frac{d}{dx}(3x^4-5x^2+7)?

Please select an answer first.

Correct answer: B

The correct answer is 12x3−10x12x^3-10x.

  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 each term: 12x^3-10x.
  3. Interpret the result: The correct answer is 12x3−10x12x^3-10x.

Differentiate each term: 12x^3-10x.

3. QuestionMultiple choice

What is ddx(tan⁡x)\frac{d}{dx}(\tan x)?

Please select an answer first.

Correct answer: C

The correct answer is sec⁡2x\sec^2 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: The derivative of tangent is secant squared.
  3. Interpret the result: The correct answer is sec⁡2x\sec^2 x.

The derivative of tangent is secant squared.

4. QuestionMultiple choice

What is ddx(ln⁡(5x))\frac{d}{dx}(\ln(5x))?

Please select an answer first.

Correct answer: D

The correct answer is 1x\frac{1}{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: By the chain rule, (1/(5x))·5=1/x.
  3. Interpret the result: The correct answer is 1x\frac{1}{x}.

By the chain rule, (1/(5x))·5=1/x.

5. QuestionMultiple choice

What is ddx(x2ex)\frac{d}{dx}(x^2e^x)?

Please select an answer first.

Correct answer: A

The correct answer is ex(x2+2x)e^x(x^2+2x).

  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 product rule: 2xe^x+x^2e^x=e^x(x^2+2x).
  3. Interpret the result: The correct answer is ex(x2+2x)e^x(x^2+2x).

Use the product rule: 2xe^x+x^2e^x=e^x(x^2+2x).

6. QuestionMultiple choice

What is ddx(x+1x−2)\frac{d}{dx}\left(\frac{x+1}{x-2}\right)?

Please select an answer first.

Correct answer: B

The correct answer is −3(x−2)2-\frac{3}{(x-2)^2}.

  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 quotient rule gives [(x-2)-(x+1)]/(x-2)^2=-3/(x-2)^2.
  3. Interpret the result: The correct answer is −3(x−2)2-\frac{3}{(x-2)^2}.

The quotient rule gives [(x-2)-(x+1)]/(x-2)^2=-3/(x-2)^2.

7. QuestionMultiple choice

What is ddx(3x+1)\frac{d}{dx}(\sqrt{3x+1})?

Please select an answer first.

Correct answer: C

The correct answer is 323x+1\frac{3}{2\sqrt{3x+1}}.

  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: Rewrite as (3x+1)^(1/2) and apply the chain rule.
  3. Interpret the result: The correct answer is 323x+1\frac{3}{2\sqrt{3x+1}}.

Rewrite as (3x+1)^(1/2) and apply the chain rule.

8. QuestionMultiple choice

What is ddx(sin⁡(4x))\frac{d}{dx}(\sin(4x))?

Please select an answer first.

Correct answer: D

The correct answer is 4cos⁡(4x)4\cos(4x).

  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 inner derivative is 4, so multiply cos(4x) by 4.
  3. Interpret the result: The correct answer is 4cos⁡(4x)4\cos(4x).

The inner derivative is 4, so multiply cos(4x) by 4.

9. QuestionMultiple choice

What is f′(1)f'(1) for f(x)=x3+2x2f(x)=x^3+2x^2?

Please select an answer first.

Correct answer: A

The correct answer is 7.

  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: f'(x)=3x^2+4x, and f'(1)=3+4=7.
  3. Interpret the result: The correct answer is 7.

f'(x)=3x^2+4x, and f'(1)=3+4=7.

10. QuestionMultiple choice

What is the derivative of f(x)=e2x−1f(x)=e^{2x-1}?

Please select an answer first.

Correct answer: B

The correct answer is 2e2x−12e^{2x-1}.

  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 chain rule multiplies e^(2x-1) by the inner derivative 2.
  3. Interpret the result: The correct answer is 2e2x−12e^{2x-1}.

The chain rule multiplies e^(2x-1) by the inner derivative 2.

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