Differentiate f(x) = x⁴.
- Bring down the exponent 4.
- Reduce the exponent to 3.
Answer: f′(x) = 4x³
Basic Differentiation Rules Worksheets
Differentiate functions exactly, state necessary domains and check the resulting symbolic expression.
Learn the skill
Differentiating polynomials with the power rule
A derivative gives an instantaneous rate of change. For a power x^n, the power rule is d/dx(x^n) = n x^(n−1), wherever the rule applies. Differentiate a polynomial term by term, keeping each coefficient.
A method you can use
- Multiply each coefficient by its exponent.
- Reduce that exponent by one; the derivative of a constant is zero.
- Combine the derivatives of the terms, then substitute a point only if a numerical slope is requested.
Worked examples
Differentiate g(x) = 3x² − 5x + 7.
- The derivative of 3x² is 6x; the derivative of −5x is −5.
- The constant 7 contributes zero.
Answer: g′(x) = 6x − 5
Find the slope of y = x³ at x = 2.
- The derivative is 3x².
- Evaluate at 2: 3 × 4 = 12.
Answer: 12
Try it yourself
Solve these before opening the answers. The printable sets below give you more practice.
1. Differentiate f(x) = 2x³ + 4x − 9.
Show answer and explanation
f′(x) = 6x² + 4
Apply the power rule to each term and drop the constant.
2. Find the slope of y = x² at x = 5.
Show answer and explanation
10
The derivative is 2x; at x = 5 it is 10.
Before you begin
Manipulate algebraic expressions and understand functions, limits and derivative notation.
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
Differentiate with respect to x: f(x) = 2·x^4+4·x.
2
Differentiate with respect to x: f(x) = 1·x^5+4·x.
3
Differentiate with respect to x: f(x) = 1·x^3+6·x.
4
Differentiate with respect to x: f(x) = 5·x^2+8·x.
5
Differentiate with respect to x: f(x) = 7·x^4+3·x.
6
Differentiate with respect to x: f(x) = 2·x^2+1·x.
Answer Key
1. f′(x) = 8·x^3+42. f′(x) = 5·x^4+43. f′(x) = 3·x^2+64. f′(x) = 10·x^1+85. f′(x) = 28·x^3+36. f′(x) = 4·x^1+1