Free printable worksheet

Applications of Derivatives Worksheet

Twelve derivative applications involving extrema, intervals, concavity, motion, and optimization.

CalculusGrades 11-124 pagesAnswer key included

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What this worksheet includes

  • Twelve derivative applications involving extrema, intervals, concavity, motion, and optimization.
  • 4-page printable PDF with a complete answer key
  • Designed for Grades 11-12 calculus practice
  • First-page preview available before the free email unlock

How to use it

Use this applications of derivatives worksheet for guided practice, independent review, homework, tutoring sessions, or a quick skills check. The spacious layout is designed to leave room for students to show their work rather than guess from crowded answer choices.

After students finish, compare their reasoning with the included answer key. When an answer is incorrect, have the student locate the first step where their work changed direction before trying the problem again.

Who this practice is for

This resource is designed for grades 11-12 students working on calculus. Teachers, tutors, homeschool families, and parents can preview the first page before deciding whether the level and spacing fit the student.

Looking for a broader sequence? Browse all Calculus worksheets.

Topic guide

How to approach applications of derivatives

Applications of Derivatives practice is most effective when students identify the governing definition, theorem, or rule before beginning a calculation. Twelve derivative applications involving extrema, intervals, concavity, motion, and optimization.

This calculus set moves from direct applications into mixed review. Students should preserve exact values, label each important step, and use the answer key to locate the first point where an incorrect solution changes direction.

Learning objectives

  • Recognize the central rules and representations used in applications of derivatives
  • Apply an efficient procedure to solve applications of derivatives problems
  • Check results using notation, substitution, estimation, or a relevant inverse process

Skills to review first

  • Core vocabulary connected to applications of derivatives
  • Accurate arithmetic and algebraic simplification
  • Reading mathematical notation and showing work in a logical sequence
Worked example

Follow the reasoning step by step

Plan and solve a representative applications of derivatives problem from Part A.

  1. List the given information and state what must be found.
  2. Select the definition, theorem, formula, or procedure that applies to applications of derivatives.
  3. Substitute or transform carefully, keeping one logical change on each line.
  4. Check the result against the original conditions and include units or restrictions when needed.

Solution: A complete solution includes the correct result and enough reasoning to verify it.

Try before downloading

Three sample problems

Work each problem, then open the answer to check your result.

Problem 1 · foundations

For f(x)=x2−6x+5f(x)=x^2-6x+5, find all critical numbers.

Show answer

x=3x=3

Problem 2 · skill practice

For f(x)=−x2+8x−3f(x)=-x^2+8x-3, find the absolute maximum value and the xx-value where it occurs.

Show answer

13 at x=413\text{ at }x=4

Problem 3 · mixed review

The position of an object is s(t)=t3−6t2+9ts(t)=t^3-6t^2+9t. Find its velocity at t=2t=2.

Show answer

−3-3

Error check

Common mistakes to watch for

  • Beginning a calculation before identifying the relevant rule or given information
  • Dropping a sign, unit, restriction, or exact-value symbol during simplification
  • Comparing only the final answer instead of checking the first incorrect step
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