Power Rule
Derivative of xⁿ.
Key idea
Derivative of xⁿ.
In depth
d/dx xⁿ = n xⁿ⁻¹. It follows from the binomial expansion in the limit definition. It works for any real n, including negatives and fractions.
Worked example
d/dx x⁵ = 5x⁴. d/dx √x = d/dx x^(1/2) = (1/2)x^(−1/2). d/dx (1/x²) = −2x⁻³.
Formula
Real-world uses
- Finding how fast the area of a square grows as its side grows
- Quick differentiation in physics
Common mistake
Forgetting to rewrite roots and reciprocals as powers first.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Power Rule?
Derivative of xⁿ. It is a topic at the advanced level in Calculus.
What is the formula for Power Rule?
The key formula is d/dx xⁿ = n xⁿ⁻¹.
Where is Power Rule used in real life?
Real-world examples: Finding how fast the area of a square grows as its side grows; Quick differentiation in physics.
What should I learn before Power Rule?
Make sure you are comfortable with Derivative as Velocity, the previous topic in Calculus.
What is a common mistake with Power Rule?
Forgetting to rewrite roots and reciprocals as powers first.
How can I practise Power Rule?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Power Rule?
Move on to Constant and Sum Rules next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.