Skip to content
MathAnvil
§ Calculus

Differentiation Worksheets

Free PDF · Problems + answer key · Instant download

Easy

10 problems

Medium

20 problems

Hard

20 problems

Mixed

30 problems

Free printable differentiation worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from power rule: d/dx[ax^n] at the easy level through to chain rule: d/dx[(ax+b)^n] at the advanced level.

R1R2CCSS.HSF.IF

What is differentiation?

Differentiation is the mathematical process of finding the derivative of a function, which represents the instantaneous rate of change at any given point. The derivative of f(x) = 3x² is f'(x) = 6x, meaning at x = 2, the function changes at a rate of 12 units per unit input. This fundamental operation in calculus transforms polynomials, trigonometric functions, and exponentials into expressions that describe their slopes.

Why it matters

Differentiation appears throughout physics, engineering, and economics to model changing quantities. In physics, differentiating position functions gives velocity — if an object's position is s(t) = 16t², its velocity is s'(t) = 32t feet per second. Engineers use derivatives to optimize bridge designs and minimize material costs. In economics, marginal cost functions derive from total cost functions through differentiation — if producing x items costs C(x) = 0.01x³ + 2x + 100 dollars, the marginal cost is C'(x) = 0.03x² + 2 dollars per additional item. Calculus courses build on differentiation to explore integration, differential equations, and multivariable calculus, making it essential preparation for advanced STEM fields.

Common mistakes to watch for

  • ✗A common error with the power rule is writing the derivative of x³ as 3x³ instead of 3x², forgetting to reduce the exponent by 1 after bringing it down as a coefficient.
  • ✗When differentiating 2x⁴ + 5x, a frequent mistake is writing 8x⁴ + 5 instead of 8x³ + 5, applying the power rule incorrectly to the linear term.
  • ✗With the chain rule, differentiating (3x + 1)² often produces 2(3x + 1) instead of 6(3x + 1), missing the derivative of the inner function 3x + 1.

Questions teachers ask

What is the difference between a derivative and differentiation?+
Differentiation is the process or operation of finding a derivative, while a derivative is the result — the function that represents the rate of change. For f(x) = x³, differentiation produces the derivative f'(x) = 3x².
How do you check if a derivative is correct?+
Verify derivatives by checking specific points or using differentiation rules backwards. For f(x) = 2x³ with derivative f'(x) = 6x², substitute x = 2: f'(2) = 24, which should match the slope of the tangent line at that point.
Why does the derivative of a constant equal zero?+
Constants don't change, so their rate of change is zero. The graph of f(x) = 7 is a horizontal line with slope 0 everywhere, making f'(x) = 0. This follows from the power rule: 7 = 7x⁰, so the derivative is 0 × 7x⁻¹ = 0.
What does it mean when a derivative is negative?+
A negative derivative indicates the function is decreasing at that point. For f(x) = -x² + 4, the derivative f'(x) = -2x is negative when x > 0, showing the parabola slopes downward on the right side of the vertex.
When do you use the chain rule versus the power rule?+
Use the power rule for simple powers like x⁵ or 3x⁴. Use the chain rule when the base contains a function, like (2x + 3)⁴ or sin(3x). The chain rule handles composite functions where one function is inside another.
Generate worksheet →Free · No account · Unlimited

Pick a difficulty

Click any level to open the generator with that difficulty pre-selected.

Beginner

Generate →
Concepts
Power rule: d/dx[ax^n]
Range
a: 1–5, n: 2–4
Steps
1 step
Example
d/dx[3x²]

Easy

Generate →
Concepts
Differentiate a cubic polynomial
Range
a: 1–3, b: −4 to 4, c: −6 to 6
Steps
4–5 steps
Example
d/dx[2x³ + 3x² − x + 1]

Medium

Generate →
Concepts
Derivatives of sin, cos, e^x
Range
coefficient: 1–4
Steps
1 step
Example
d/dx[3 sin(x)]

Hard

Generate →
Concepts
Chain rule: d/dx[(ax+b)^n]
Range
a: 2–5, b: −4 to 4, n: 2–4
Steps
3 steps
Example
d/dx[(2x + 3)³]

Try a sample problem

Try it right now

Click “Generate a problem” to see a fresh example of this technique.

Learn the theory → Read our differentiation guide with worked examples.

Practice online → Interactive differentiation problems with instant feedback.