Power Rule
d/dx xⁿ = n·xⁿ⁻¹.
About this topic
The power rule states that the derivative of xⁿ is n·xⁿ⁻¹: bring the exponent down as a multiplier and reduce it by one. So d/dx x⁵ = 5x⁴, and d/dx √x = ½x^(−1/2). Constants differentiate to zero, and constant multiples pass through.
Combined with the sum rule, it lets you differentiate any polynomial term by term. For f(x) = 4x³ − 7x² + 2x − 9, the derivative is 12x² − 14x + 2. Rewrite roots and reciprocals as powers first: 1/x² is x⁻², with derivative −2x⁻³, and the cube root of x is x^(1/3), with derivative (1/3)x^(−2/3).
The rule gives slopes of tangent lines and velocities quickly. If a position is s = t³ − 6t, the velocity is 3t² − 6, which is zero at t = √2 when the object momentarily stops. Physics, engineering and economics use it daily.
Common errors: forgetting to lower the exponent, differentiating a constant as if it were a variable term, and applying the rule to exponentials such as 2ˣ, where the variable is the exponent and not the base. That case needs a different rule involving ln 2.
Key ideas
- Bring down the exponent and reduce it by 1
- Constants disappear
- Works for fractional exponents
Quick check
Use cases: where Power Rule helps
- Homework: review the key idea behind Power Rule before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Power Rule without notes.
- Visual learning: play with the interactive lab to see how Power Rule changes when you adjust its inputs.
- Teaching a friend: explain Power Rule in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Power Rule?
d/dx xⁿ = n·xⁿ⁻¹.
What will I learn in Power Rule?
Bring down the exponent and reduce it by 1. Constants disappear. Works for fractional exponents.
Is Power Rule part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Power Rule lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Power Rule?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.