Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC prerequisite skills and Unit 6; CCSS.HSF.IF, HSF.TF, HSA.SSE; CCSS.MP1, MP2, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-FOUND; TEKS 19 TAC section 111.55

CAL2.9 Checking Antiderivatives by Differentiation Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.9 Checking Antiderivatives by Differentiation. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.9 Checking Antiderivatives by Differentiation - Select whether a proposed antiderivative is correct by differentiating and comparing to the original integrand. Standards: AP Calculus BC prerequisite skills and Unit 6; CCSS.HSF.IF, HSF.TF, HSA.SSE; CCSS.MP1, MP2, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-FOUND; TEKS 19 TAC section 111.55

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Calculus II

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related CAL2.9 Checking Antiderivatives by Differentiation skill assessments:

Problem 1Difficulty: Mastery

Check whether the proposed function is an antiderivative. On intervals where the expressions are defined. Target derivative: f(x) = (x² - 1)/(x² + 1)². Proposed function: F(x) = x/(x² + 1). Which statement is true?

ACorrect; differentiating the proposed function gives the target derivative.

BCorrect; differentiating the proposed function gives (x² - 1)/(x² + 1)².

CCorrect; any mismatch can be fixed by adding + C to the proposed function.

DIncorrect; differentiating the proposed function gives (1 - x²)/(x² + 1)², not (x² - 1)/(x² + 1)².

Solution & Explanation:Differentiate the proposed function x/(x² + 1); the derivative is (1 - x²)/(x² + 1)². This is not the target derivative (x² - 1)/(x² + 1)², so the proposed function is not an antiderivative. Adding a constant would not change the derivative, so it cannot fix this mismatch. The quotient rule gives the negative of the target expression.

Problem 2Difficulty: Mastery

Verify the proposed antiderivative by differentiation. On intervals where the expressions are defined. Target derivative: f(x) = cos(x)/(1 + sin²(x)). Proposed function: F(x) = arctan(sin(x)). Which statement is true?

AIncorrect; constants cannot disappear when differentiating a proposed antiderivative.

BIncorrect; differentiating the proposed function gives the target derivative plus C, not the target derivative.

CCorrect; differentiating the proposed function gives cos(x)/(1 + sin²(x)), matching the target derivative.

DIncorrect; omitting + C makes the proposed single antiderivative invalid.

Solution & Explanation:Differentiate the proposed function arctan(sin(x)); the derivative is cos(x)/(1 + sin²(x)), which equals the target derivative cos(x)/(1 + sin²(x)). Therefore the proposed function is a valid antiderivative on the stated domain. A missing + C does not invalidate one proposed antiderivative; + C describes the full family. The derivative of sin(x) is cos(x), matching the numerator.

Problem 3Difficulty: Mastery

Differentiate the proposed function to test the antiderivative claim. On intervals where the expressions are defined. Target derivative: f(x) = sin(x)/(1 + cos²(x)). Proposed function: F(x) = -arctan(cos(x)). Which statement is true?

AIncorrect; the derivative must equal the proposed function, not the target derivative.

BCorrect; differentiating the proposed function gives sin(x)/(1 + cos²(x)), matching the target derivative.

CIncorrect; differentiating the proposed function gives a constant multiple of the target derivative, not the target derivative.

DIncorrect; constants cannot disappear when differentiating a proposed antiderivative.

Solution & Explanation:Differentiate the proposed function -arctan(cos(x)); the derivative is sin(x)/(1 + cos²(x)), which equals the target derivative sin(x)/(1 + cos²(x)). Therefore the proposed function is a valid antiderivative on the stated domain. A missing + C does not invalidate one proposed antiderivative; + C describes the full family. The derivative of cos(x) is -sin(x), and the leading negative cancels that sign.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Calculus II Math students. It aligns with educational standard AP Calculus BC prerequisite skills and Unit 6; CCSS.HSF.IF, HSF.TF, HSA.SSE; CCSS.MP1, MP2, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-FOUND; TEKS 19 TAC section 111.55 (Select whether a proposed antiderivative is correct by differentiating and comparing to the original integrand. Standards: AP Calculus BC prerequisite skills and Unit 6; CCSS.HSF.IF, HSF.TF, HSA.SSE; CCSS.MP1, MP2, MP5, MP6, MP7; Integrated Calculus II Standards CAL2-FOUND; TEKS 19 TAC section 111.55).