\frac{d}{dx}x^n=nx^{n-1}Variables
- n: constant exponent
- x: variable
How to use this formula
Differentiates a power of x.
Important notes
- For real exponents, domain restrictions may apply.
Quick example
d(x⁵)/dx=5x⁴.
Applicability, worked calculation, and verification
Domain and applicability
Apply the power rule on intervals where x^n is differentiable under the stated exponent and number-system assumptions.
Assumptions and domain checks
- The exponent and domain must make the original function and resulting derivative well defined.
- Constant multiples should be retained and differentiated using linearity.
Do not use this formula when
- Do not use Derivative Power Rule without checking differentiability, integrability, convergence, interval, and transform-convention requirements.
Boundary and special cases
- For n=0, the derivative of the constant x^0=1 is zero even though the unsimplified pattern contains x^{-1}.
- For negative or fractional exponents, exclude points where the original real-valued power is not defined.
Equivalent and alternative forms
- For a monomial ax^n, linearity gives d(ax^n)/dx=anx^{n-1}.
Worked example
For f(x)=3x^5, f′(x)=3·5x^4=15x^4.
- Identify the coefficient and exponent of each power term before differentiating.
- Multiply the coefficient by the original exponent for the derivative coefficient.
- Reduce the exponent by one while preserving the variable and any constant multiplier.
- Verify the derivative at a sample point with a difference quotient or a trusted symbolic calculation.
Independent verification
For a chosen value of x, compare the derivative with a small-step numerical difference quotient and confirm that the values converge as the step shrinks.
Common mistakes
- Do not subtract one from the exponent without also multiplying by the original exponent.
- Do not use the elementary real-valued rule across points where a fractional or negative power is undefined.
Continue the workflow
Use Derivative Power Rule in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\frac{d}{dx}x^n=nx^{n-1}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked example, boundary cases, and independent verification method for Derivative Power Rule.
Verified against: OpenStax Calculus Volume 1
Automated quality check: Passed the core formula indexing gate.
Formula references
- Calculus Volume 1OpenStax, Rice University — Reviewed limits, derivatives, integrals, and foundational calculus formulas.
Frequently asked questions
What is the Derivative Power Rule used for?
Differentiates a power of x.
Can I copy this formula as LaTeX?
Yes. Copy \frac{d}{dx}x^n=nx^{n-1} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.