Try the rule calculator
Definition and meaning
The power rule differentiates a constant power of a variable: if f(x)=x^n, then f'(x)=n x^(n-1). The exponent may be positive, negative, fractional, or zero wherever the original expression is defined. For a polynomial, apply the rule to each term and use the constant-multiple and sum rules.
Why the rule works
The rule follows from the limit definition. Expanding (x+h)^n with the binomial theorem, subtracting x^n, and dividing by h leaves n x^(n-1) plus terms that still contain h. Those remaining terms approach zero as h approaches zero. For noninteger exponents, the same formula follows on the appropriate domain through logarithms or the generalized binomial theorem.
A reliable hand method
First identify every term whose variable is raised to a constant power. Keep its coefficient, multiply that coefficient by the exponent, and reduce the exponent by one. Differentiate constants to zero. Rewrite radicals and reciprocals as powers before working when that makes the exponent visible.
Example 1: x^7
Answer: 7x^6
- Bring 7 down as a coefficient.
- Reduce the exponent from 7 to 6.
Example 2: 4x^3-5x+9
Answer: 12x^2-5
- Differentiate each term separately.
- The constant 9 contributes zero.
Example 3: 1/x^3
Answer: -3/x^4
- Rewrite 1/x^3 as x^-3.
- Multiply by -3 and reduce the exponent to -4.
Example 4: sqrt(x)
Answer: 1/(2sqrt(x))
- Rewrite sqrt(x) as x^(1/2).
- The result is (1/2)x^(-1/2), then rewrite with a radical.
Common mistakes
A common error is subtracting one from the exponent without multiplying by the old exponent. Another is treating a coefficient as an exponent. With negative powers, students sometimes change the sign twice; x^-3 differentiates to -3x^-4. For roots, remember that the derivative formula can have a smaller domain than an informal radical rewrite suggests.
How to verify an answer
Differentiate the expression independently, compare the structure of each term, and substitute a safe numerical value into both the symbolic derivative and a centered difference quotient. Avoid points outside the original domain and points near a discontinuity. A matching numerical value increases confidence, but it cannot establish equality for every input. Algebraic reasoning and domain analysis remain necessary.
When the calculator and handwritten work differ, first check parentheses, multiplication symbols, the selected variable, and any missing chain-rule factor. Then factor or expand both answers to see whether they are equivalent. The DerivCalc methodology explains why simplification is deliberately conservative.
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Questions about this calculator
Can I use the Power Rule Calculator for homework checks?
Yes. Compare the result and named rules with your own work, then check domain restrictions. The numerical spot-check is a consistency test, not a formal proof.
Why can an equivalent answer look different?
Factoring, expanding, reciprocal notation, and trigonometric identities can produce different-looking expressions with the same values on their shared domain.
Does the calculator replace learning the rule?
No. It is designed to expose the rule and intermediate reasoning. The worked examples explain how to select and apply the method by hand.
Editorial standard: formulas and notation are checked against OpenStax Calculus Volume 1, Chapter 3. Content is written and maintained by SilverCodeLabs. Report a correction with the exact expression and expected result.