Definition of the Derivative Worksheets
Differentiate functions exactly, state necessary domains and check the resulting symbolic expression.
Before you begin
Manipulate algebraic expressions and understand functions, limits and derivative notation.
Worked example
Use the limit definition of the derivative to differentiate: f(x) = 4·x^2+7.
- Write the difference quotient [f(x + h) − f(x)]/h.
- Expand, cancel, and divide by h for h ≠ 0.
- Take the limit as h approaches zero.
Answer: f′(x) = 8·x
Check your result using the method in the example before moving to the next question.
Choose your level
These sets use the same exercise types and number ranges across level labels. Choose a fresh set for additional practice.
- Easy: Start with the preview and follow the worked example.
- Medium: Practice the same exercise types with a fresh numbered set.
- Hard: Use another set to practice explaining and checking your answers.
Questions and answer calculations checked on 2026-09-17. How we check worksheets. Free for personal and classroom use.
Preview and print an easy set
Try six questions below, or choose a full set at your level.
1
Use the limit definition of the derivative to differentiate: f(x) = 2·x^2+4.
2
Use the limit definition of the derivative to differentiate: f(x) = 1·x^2+4.
3
Use the limit definition of the derivative to differentiate: f(x) = 1·x^2+6.
4
Use the limit definition of the derivative to differentiate: f(x) = 5·x^2+8.
5
Use the limit definition of the derivative to differentiate: f(x) = 7·x^2+3.
6
Use the limit definition of the derivative to differentiate: f(x) = 2·x^2+1.
Answer Key
1. f′(x) = 4·x2. f′(x) = 2·x3. f′(x) = 2·x4. f′(x) = 10·x5. f′(x) = 14·x6. f′(x) = 4·x