10th Grade Calculus Basics Practice: Questions, Answers and Tips

Looking for 10th grade calculus basics practice? Below are worked examples, the mistakes students make most, and practice questions with an answer key.

Every example below comes from BAMS Academy's Calculus Basics practice, the same questions students answer on the site.

Calculus Basics in 10th grade, skill by skill

Limits, derivatives, and an intro to integrals. BAMS Academy breaks the unit into the skills below (part of all 10th grade math topics). Each comes with one of its practice questions.

Apply the power rule to find and evaluate derivatives

If f(x) = 1x^3, what is f'(1) (the derivative evaluated at x = 1)? (Choices: 3; 4; 6; 1.)

Answer: 3. f'(x) = 3x^2. f'(1) = 3(1)^2 = 3.

What is the derivative of 3x^3? (Choices: 3x^2; 6x^2; 9x^3; 9x^2.)

Answer: 9x^2. Using the power rule, d/dx[3x^3] = 3 × 3 × x^(3−1) = 9x^2.

Evaluate limits of polynomial and simple rational functions

What is the limit of 5x^2 + 1x + 7 as x approaches 1? (Choices: 11; 6; 13; 14.)

Answer: 13. Since this is a polynomial (continuous everywhere), the limit equals direct substitution: 5(1)² + 1(1) + 7 = 13.

What is the limit of 5x + -11 as x approaches 4? (Choices: 9; -11; -31; 20.)

Answer: 9. Since this is a linear (continuous) function, the limit equals direct substitution: 5(4) + -11 = 9.

Apply the power rule for indefinite integrals

Using the power rule for integration, ∫7x^4 dx = (coefficient)x^5 + C. What is the coefficient (as a decimal, rounded to 3 places if needed)?

Answer: 1.4. ∫7x^4 dx = 7/(4+1) × x^5 + C, so the coefficient is 7/5 = 1.4.

Using the power rule for integration, ∫7x^1 dx = (coefficient)x^2 + C. What is the coefficient (as a decimal, rounded to 3 places if needed)?

Answer: 3.5. ∫7x^1 dx = 7/(1+1) × x^2 + C, so the coefficient is 7/2 = 3.5.

Worked examples

Example 1

Question: If f(x) = 2x^3, what is f'(4) (the derivative evaluated at x = 4)?

1. 6
2. 128
3. 384
4. 96

How to solve it: f'(x) = 6x^2. f'(4) = 6(4)^2 = 96.

Answer: 96

Why the other choices don't work:

• "6" is a tempting wrong answer — found the derivative expression but forgot to evaluate it at x.
• A student who picks "128" usually evaluated the original function instead of its derivative.
• A student who picks "384" usually forgot to reduce the exponent by one before substituting.

Example 2

Question: What is the derivative of 5x^2?

1. 5x^1
2. 7x^1
3. 10x^2
4. 10x^1

How to solve it: Using the power rule, d/dx[5x^2] = 5 × 2 × x^(2−1) = 10x^1.

Answer: 10x^1

Why the other choices don't work:

• A student who picks "5x^1" usually forgot to multiply the coefficient by the original exponent.
• A student who picks "7x^1" usually added the exponent to the coefficient instead of multiplying.
• A student who picks "10x^2" usually forgot to reduce the exponent by one.

Common mistakes to watch for

• Question: What is the limit of 5x^2 + 1x + 7 as x approaches 1? (Choices: 11; 6; 13; 14.)

  Answer: 13.

  "11" is a tempting wrong answer — sign error substituting a negative x value.

• Question: What is the limit of 5x + -11 as x approaches 4? (Choices: 9; -11; -31; 20.)

  Answer: 9.

  A student who picks "-11" usually reported only the constant term, ignoring the x term.

Try these practice questions

1. If f(x) = 5x^2, what is f'(1) (the derivative evaluated at x = 1)?

   1. 8
   2. 10
   3. 11
   4. 5

2. Type the answer: what is the derivative of 2x^3, expressed as the coefficient of x^2?

3. What is the derivative of 3x^4?

   1. 7x^3
   2. 3x^3
   3. 12x^4
   4. 12x^3

4. If f(x) = 1x^3, what is f'(2) (the derivative evaluated at x = 2)?

   1. 8
   2. 24
   3. 3
   4. 12

5. Type the answer: what is the derivative of 1x^2, expressed as the coefficient of x^1?

6. What is the derivative of 5x^4?

   1. 5x^3
   2. 20x^3
   3. 20x^4
   4. 9x^3

Answer key

1. 10 — f'(x) = 10x^1. f'(1) = 10(1)^1 = 10.
2. 6 — Using the power rule, d/dx[2x^3] = 2 × 3 × x^2, so the coefficient is 6.
3. 12x^3 — Using the power rule, d/dx[3x^4] = 3 × 4 × x^(4−1) = 12x^3.
4. 12 — f'(x) = 3x^2. f'(2) = 3(2)^2 = 12.
5. 2 — Using the power rule, d/dx[1x^2] = 1 × 2 × x^1, so the coefficient is 2.
6. 20x^3 — Using the power rule, d/dx[5x^4] = 5 × 4 × x^(4−1) = 20x^3.

Keep practicing

The fastest way to build confidence is a few questions at a time, with feedback on each answer. Start with Calculus Basics practice for grade 10, where every question comes with a step-by-step explanation.

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• All 10th grade math topics
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• BAMS Academy math practice by grade
• Every 10th grade subject