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Calculus / Calculus 1 · Derivatives

Power Rule for Derivatives

The power rule is the first real shortcut in calculus. To differentiate x to any power, bring the exponent down in front and subtract one from it. Together with the constant and sum rules, it lets you differentiate any polynomial in a single line, with no limits required.

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Chapter 3 · Lesson 3 of 11Calculus 1 · Derivatives
  1. 1.Limit Definition of the Derivative
  2. 2.Differentiability
  3. 3.Power Rule for Derivatives
  4. 4.Product Rule for Derivatives
  5. 5.Quotient Rule for Derivatives
  6. 6.Chain Rule
  7. 7.Derivatives of Trig Functions
  8. 8.Derivatives of Exponential and Logarithmic Functions
  9. 9.Derivatives of Inverse Trig Functions
  10. 10.Implicit Differentiation
  11. 11.Logarithmic Differentiation
See all calculus chapters →

In the last two lessons, every derivative came from the limit definition: expand, cancel, divide by , take the limit. It works, but it’s slow, and a pattern kept showing up in the answers. became , became , and became . That pattern is the power rule, and once you know it, those page-long calculations take one line.

The power rule: the derivative with respect to x of x to the n equals n times x to the n minus 1
The power rule for derivatives

What Is the Power Rule?

For any real number ,

In words: bring the exponent down in front, then subtract one from the exponent. That’s the whole rule. The notation just means “the derivative with respect to of”, so you’ll also see it written as when .

The rule applies to a power function, where the base is the variable and the exponent is a fixed number. It doesn’t matter whether is a whole number, a negative number, a fraction or even an irrational number like .

How to Use the Power Rule

Three examples of the power rule. x to the 5 becomes 5 x to the 4. x to the negative 2 becomes negative 2 x to the negative 3. x to the one half becomes one half x to the negative one half. In each case the exponent moves down in front and the new exponent is one less
Bring the exponent down, then subtract 1, whatever kind of number it is
  1. Write the term as a power of . Roots and fractions need rewriting first (more on that below).
  2. Bring the exponent down and multiply it by any coefficient already there.
  3. Subtract 1 from the exponent. With negatives and fractions, this is where the arithmetic goes wrong, so take it slowly: , not .
  4. Tidy up, usually by turning negative and fractional exponents back into fractions and roots.

Here’s the power rule applied to a range of powers:

FunctionDerivativeNote
, so
any real exponent works

Rewriting Roots and Fractions First

Table: square root of x is x to the one half, derivative one half x to the negative one half, which simplifies to 1 over 2 root x. 1 over x cubed is x to the negative 3, derivative negative 3 x to the negative 4, or negative 3 over x to the 4. 1 over root x is x to the negative one half, derivative negative one half x to the negative three halves. Cube root of x squared is x to the two thirds, derivative two thirds x to the negative one third, or 2 over 3 cube root x
Most power rule mistakes happen before the rule is even used

The power rule only sees , so anything written with a root sign or with in a denominator has to be converted first. Two facts from algebra do all the work:

So the derivative of is . It’s the same answer the limit definition gave in the previous lesson, with none of the conjugate work.

The Constant, Constant Multiple and Sum Rules

The power rule handles single powers of . Three small rules extend it to whole polynomials:

RuleIn symbolsExample
Constant rule
Constant multiple rule
Sum and difference rule

Each has a simple picture. A constant’s graph is a flat line, so its slope is 0. Multiplying a function by 5 stretches its graph vertically by 5, so every slope is 5 times steeper. And when you add two functions, their rates of change add too. In practice this means you can differentiate a polynomial term by term, with constants dropping out and coefficients coming along for the ride.

What the Derivative Tells You on a Graph

Top: the graph of x cubed with tangent lines at x = negative 1, 0 and 1, with slopes 3, 0 and 3. Bottom: the graph of 3x squared with points at negative 1, 0 and 1 at heights 3, 0 and 3, matching the slopes above
The height of f′ at each x is the slope of f there

The power rule gives a whole new function, not just a number. For , it gives , and each value of is a slope on the graph of . At , and the cubic is momentarily flat. At , and the curve climbs steeply. Because is never negative, never goes downhill, which you can confirm from its graph.

Why the Power Rule Works

Proof in four steps. Expand x plus h to the n: x to the n plus n x to the n minus 1 times h plus terms with h squared. Subtract x to the n. Divide by h: n x to the n minus 1 plus terms with h. Let h approach 0: every h term vanishes, leaving n x to the n minus 1
The derivation of the power rule from the limit definition

Here’s the derivation for a positive whole-number exponent . Start from the definition:

The binomial theorem expands as . Subtracting removes the first term. Dividing by leaves plus terms that each still contain at least one . As , those all disappear, and is what’s left.

For negative, fractional and irrational exponents the rule is still true, but the proof needs tools from later in this chapter, logarithmic and implicit differentiation. In the meantime you’ve already checked two of those cases by hand: gave and gave .

Example 1: A Polynomial

Differentiate .

Term by term: , then , then , and the constant contributes 0.

Example 2: Roots and Fractions

Differentiate .

Rewrite first: . Then

Watch the sign on the second term: times is .

Example 3: Multiply Out Before Differentiating

Differentiate .

The derivative of a product is not the product of the derivatives. You’ll learn the product rule soon, but for now, expand first: , so

Example 4: Split a Fraction

Differentiate .

When the denominator is a single term, divide it into each term of the numerator: . Then

Example 5: The Equation of a Tangent Line

Find the tangent line to at .

The point is . The slope is at , which is . Point-slope form gives , so the tangent line is

Example 6: Where Is the Tangent Horizontal?

Graph of y equals x cubed minus 12x with dashed horizontal tangent lines at the turning points (negative 2, 16) and (2, negative 16), where the derivative 3x squared minus 12 equals zero
Horizontal tangents occur where the derivative is zero

Find the points where has a horizontal tangent line.

A horizontal line has slope 0, so solve : , so and . The points are and , which are exactly the turning points on the graph. Setting the derivative equal to zero to find peaks and valleys is one of the main jobs of derivatives, and it comes back in the applications chapter.

Example 7: Velocity From Position

A particle moves along a line with position metres after seconds. How fast is it moving at ?

Velocity is the derivative of position: . At , . The particle is moving at 3 metres per second, and the minus sign says it’s moving backwards, toward smaller .

Common Mistakes With the Power Rule

  • Subtracting 1 the wrong way with negative exponents. , not . One less than is .
  • Differentiating a root without rewriting it. Convert to and to before using the rule.
  • Giving a constant a nonzero derivative. , and in the derivative is 8, not or 0.
  • Using the power rule on exponentials. and have the variable in the exponent, so the power rule doesn’t apply.
  • Differentiating products and quotients factor by factor. , not . Simplify first, or use the product and quotient rules.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Differentiate .

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.

Answer:

Problem 2. Differentiate .

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, so the derivative is .

Answer:

Problem 3. Differentiate .

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gives .

Answer:

Problem 4. If , find .

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, so .

Answer: 7

Problem 5. Find the tangent line to at .

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The point is and the slope is , so .

Answer:

The power rule handles sums of powers, but not products like or quotients like . Those need the product and quotient rules, coming next.

Frequently Asked Questions

What is the power rule?+

The power rule says that the derivative of is : . You bring the exponent down as a coefficient and lower the exponent by one. For example, the derivative of is .

Does the power rule work for negative and fractional exponents?+

Yes. It works for every real exponent. and . Roots and fractions just have to be rewritten as powers of first: and .

Why does the power rule work?+

For a positive whole number , expand with the binomial theorem inside the limit definition of the derivative. After subtracting and dividing by , every term except still contains an , so they all vanish as . The rule for other exponents is proved later with implicit or logarithmic differentiation.

What is the derivative of a constant?+

Zero. A constant function such as has a horizontal graph, so its slope is 0 everywhere. This matches the power rule, since and .

Can you use the power rule on e^x or 2^x?+

No. The power rule is for a variable base and a constant exponent, like . In and the exponent is the variable, so they're exponential functions with their own rule: . Writing is a common mistake.

What is the general power rule?+

The general (or generalized) power rule handles a function raised to a power: , where is a function of . For example, . It's the power rule combined with the chain rule, covered later in this chapter.

Related lessons

Limit Definition of the DerivativeDifferentiabilityEvaluating LimitsL'Hôpital's Rule