The power rule is the shortcut for the derivative of a power of x: if f(x) = xn, then f′(x) = nxn − 1. Bring the exponent down as a coefficient and subtract 1 from it, so x5 becomes 5x4. It works for any real exponent, including negative and fractional ones.
In Spanish
regla de la potencia
Also called
power rule for derivatives, power rule of differentiation, exponent rule for derivatives
Grade level
Calculus
CalculusAP
Power Rule in a picture
The power rule turns x4 into 4x3 in two moves. On the graph of y = x2, its derivative 2x gives the slope of the tangent line at every point.
Formal definition and notation
For any real numbern, ddx( xn ) = nxn − 1, wherever xn is defined and differentiable (for example x > 0 when n is not an integer). For positive integers it follows from the limit definition of the derivative and the binomial expansion of (x + h)n. Combined with the constant multiple and sum rules, it differentiates every polynomial term by term.
Symbol
Read it as
Meaning
ddx( xn )
d d x of x to the n
the derivative of xn with respect to x
f′(x)
f prime of x
the derivative of f
nxn − 1
n times x to the n minus 1
the result of the power rule
Key facts
Rewrite roots and fractions as powers first: √x = x12 and 1x2 = x−2.
Differentiate polynomials term by term: ddx(3x4 − 2x + 7) = 12x3 − 2.
The derivative of a constant is 0, because its graph is flat; the derivative of x is 1.
The power rule needs the variable in the base. It does not apply to 2x, whose derivative is 2xln 2.
For a power of an inner function, such as (3x + 1)4, use the power rule together with the chain rule.
Power rule
Power rule
ddx( xn ) = nxn − 1
n
the exponent: any real number
nxn − 1
new coefficient n, new exponent one less
With a coefficient: ddx( cxn ) = cnxn − 1. Special cases: ddx(x) = 1 and the derivative of a constant is 0.
Integration runs the rule backward: add 1 to the exponent, then divide by the new exponent. ∫x3dx = x44 + C.
Power Rulevs.Chain rule
The power rule handles xn. For (inner)n, multiply by the inner derivative too: ddx(2x + 1)3 = 3(2x + 1)2 · 2.
Quick check: 3 questions
Question 1 of 3
What is the derivative of x10?
Show answer and explanation
Answer: B. Bring down 10 and subtract 1: 10x9. 10x10 forgets to lower the exponent, 9x10 mixes the steps up, and x1111 is the antiderivative.
Question 2 of 3
What is ddx( 1x2 )?
Show answer and explanation
Answer: D. Rewrite as x−2; the derivative is −2x−3 = −2x3. −2x subtracts 1 in the wrong direction, and 2x differentiates x2 instead.
Question 3 of 3
If f(x) = x3 + 4x, what is f′(−1)?
Show answer and explanation
Answer: A.f′(x) = 3x2 + 4, so f′(−1) = 3 + 4 = 7. −5 is f(−1), the value of the function, and 1 squares −1 wrongly as −1.
Power Rule: frequently asked questions
Does the power rule work for negative and fractional exponents?
Yes. The rule ddxxn = nxn − 1 works for every real exponent. First rewrite the function as a power: 3√x = x13 has derivative 13x−23, and 5x = 5x−1 has derivative −5x−2. Just be careful subtracting 1 from a fraction or a negative number.
What is the derivative of a constant?
It is 0. A constant function such as f(x) = 8 has a horizontal graph, so its slope is 0 everywhere. You can also see it from the power rule: 8 = 8x0, and bringing down the exponent 0 gives 0 · 8x−1 = 0. That is why constants disappear when you differentiate a polynomial.
Why does the power rule work?
For x2, the limit definition gives (x + h)2 − x2h = 2xh + h2h = 2x + h, which approaches 2x as h → 0. For xn, expanding (x + h)n gives xn + nxn − 1h plus terms with h2 or higher, and only the nxn − 1 part survives the limit.
Updated November 2026·Reviewed by the Effortless Math team·Standards: n/a
Cite this entry
Effortless Math Team. “Power Rule.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/power-rule-derivatives/
Flashcard
TermPower Ruleddx( xn )
DefinitionBring the exponent down and subtract 1: ddxxn = nxn − 1. Example: the derivative of x4 is 4x3.
DerivativeA derivative measures how fast a function’s output changes at one exact input: its instantaneous rate of change. On a graph it is the slope of the tangent line at that point. The derivative of f(x) = x2 is f′(x) = 2x, so at x = 3 the curve is rising with slope 6.Calculus · CalculusReal NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1Positive IntegerA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Problem Solving · Grade 6, Grade 7, Algebra 1BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7Chain RuleThe chain rule is the rule for the derivative of a composite function: if y = f(g(x)), then y′ = f′(g(x)) · g′(x). Differentiate the outer function, leave the inner function inside, then multiply by the derivative of the inner function. For (x2 + 1)3, the derivative is 3(x2 + 1)2 · 2x.Calculus · CalculusSimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Negative ExponentA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a−n = 1an for a ≠ 0. So 2−3 = 18 and 10−2 = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Algebra · Grade 8, Algebra 1Exponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2, PrecalculusLaws of ExponentsThe laws of exponents are the rules for simplifying powers that share a base or an exponent. Multiply powers of the same base by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them: x2 · x5 = x7, x6x2 = x4, (x3)2 = x6.Algebra · Grade 8, Algebra 1, Algebra 2IntegralAn integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral ∫baf(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: ∫ 2xdx = x2 + C.Calculus · CalculusLimitA limit is the value a function gets closer and closer to as the input approaches a certain number. We write limx → af(x) = L. The function need not even be defined at a. For f(x) = x2 − 1x − 1, f(1) is undefined, but the limit as x → 1 is 2.Calculus · Precalculus, CalculusRational ExponentA rational exponent is an exponent written as a fraction, and it stands for a root. The denominator is the root and the numerator is the power: a1n = n√a and amn = ( n√a )m. For example, 912 = 3 and 2723 = 32 = 9.Algebra · Algebra 1, Algebra 2Critical PointA critical point of a function is an input x = c in its domain where the derivative is 0 or does not exist. These are the only places a smooth graph can turn around, so every local maximum or minimum happens at a critical point. For f(x) = x2 − 4x, f′(x) = 2x − 4 = 0 at x = 2.Calculus · Calculus