Power Rule in Calculus: Formula, Proof, Examples

What Is The Power Rule?

The Power Rule is a calculus rule for finding the derivative of a power function, a function of the form $x^n$ where $n$ is a fixed real number. It states:

$$\frac{d}{dx}\left[x^n\right] = n,x^{n-1}$$

Read it as a two-step recipe: bring the exponent down as a multiplier, then subtract one from the exponent. So $\frac{d}{dx}\left[x^7\right] = 7x^6$, and $\frac{d}{dx}\left[x^3\right] = 3x^2$.

Two clarifications matter before anything else, because this exact name is shared by a different rule:

• This is the calculus Power Rule, not the algebra one. In algebra, the "power of a power" rule $\left(x^a\right)^b = x^{ab}$ is about simplifying exponents. The calculus Power Rule here is about differentiating, finding the derivative that measures an instantaneous rate of change. The two are different tools that happen to share a word. For the algebra version, see exponent rules.

• The base must be the variable and the exponent must be constant. The Power Rule applies to $x^n$ (variable base, fixed power). It does not apply to $a^x$ (fixed base, variable exponent) such as $2^x$ or $e^x$, which are exponential functions with their own rule.

Geometrically, the derivative $\frac{d}{dx}\left[x^n\right]$ is the slope of the tangent line to the curve $y = x^n$ at each point. The Power Rule is really a formula for how steep that curve is everywhere at once. This slope reading is the thread that runs through all of calculus.

What Is The Power Rule Formula?

The formula, stated with its full generality, is:

$$\frac{d}{dx}\left[x^n\right] = n,x^{n-1}, \qquad n \in \mathbb{R}$$

Paired with two companion rules, it differentiates almost any polynomial or root expression you meet:

• Constant multiple rule: $\frac{d}{dx}\left[c,f(x)\right] = c,\frac{d}{dx}\left[f(x)\right]$. A constant coefficient rides along untouched, so $\frac{d}{dx}\left[5x^3\right] = 5 \cdot 3x^2 = 15x^2$.

• Sum rule: $\frac{d}{dx}\left[f(x) + g(x)\right] = \frac{d}{dx}\left[f(x)\right] + \frac{d}{dx}\left[g(x)\right]$. Differentiate a polynomial term by term.

One special case is worth naming on its own: a constant. Since any constant equals $x^0$, the rule gives $\frac{d}{dx}\left[x^0\right] = 0 \cdot x^{-1} = 0$. A constant does not change, so its rate of change is zero, and the Power Rule agrees.

Table: The Power Rule applied across values of $n$.

How Do You Prove The Power Rule?

For a positive integer $n$, the Power Rule falls straight out of the limit definition of the derivative and the binomial expansion of $(x+h)^n$.

Start from first principles:

$$\frac{d}{dx}\left[x^n\right] = \lim_{h \to 0} \frac{(x+h)^n - x^n}{h}$$

Expand $(x+h)^n$ with the binomial theorem. Writing the binomial coefficient $\binom{n}{2} = \tfrac{n(n-1)}{2}$ so no factorial symbol is needed:

$$(x+h)^n = x^n + n,x^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots + h^n$$

Subtract $x^n$ and the leading term cancels:

$$(x+h)^n - x^n = n,x^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots + h^n$$

Now divide by $h$. Every remaining term has at least one factor of $h$, so dividing removes exactly one:

$$\frac{(x+h)^n - x^n}{h} = n,x^{n-1} + \binom{n}{2}x^{n-2}h + \cdots + h^{n-1}$$

Take the limit as $h \to 0$. Every term still carrying an $h$ vanishes, and only the first term survives:

$$\frac{d}{dx}\left[x^n\right] = \lim_{h \to 0}\left[,n,x^{n-1} + \binom{n}{2}x^{n-2}h + \cdots + h^{n-1}\right] = n,x^{n-1}$$

That single surviving term, $n,x^{n-1}$, is the whole rule. The binomial expansion hands you exactly one term without a leftover $h$, and that term is the derivative.

The proof above assumes $n$ is a positive integer, because the binomial theorem in this form needs a whole-number power. The rule still holds for negative and fractional $n$, by two other routes:

• Negative integers ($n = -m$): write $x^n = \dfrac{1}{x^m}$ and differentiate with the quotient (reciprocal) rule. The result simplifies to $-m,x^{-m-1} = n,x^{n-1}$, the same formula. This is why the rule extends cleanly to negative exponents.

• Rational powers ($n = \tfrac{p}{q}$): set $y = x^{p/q}$, raise both sides to the power $q$ to get $y^q = x^p$, and differentiate implicitly. Solving for $\frac{dy}{dx}$ returns $\frac{p}{q},x^{(p/q)-1}$, again the Power Rule. This covers roots and other fractional exponents.

For any real exponent, rewriting $x^n = e^{n \ln x}$ and applying the chain rule extends the rule to all of $\mathbb{R}$, which is why the formula is quoted for every real $n$.

How Do You Use The Power Rule? (Worked Examples)

Each example applies the two-step recipe: bring the exponent down, subtract one. Where the exponent is negative or fractional, the same steps still hold.

Example 1: Differentiate $x^5$.

$$\frac{d}{dx}\left[x^5\right] = 5,x^{5-1} = 5x^4$$

Final answer: $5x^4$. Quick check against the limit definition: the surviving binomial term for $n=5$ is $5x^{5-1} = 5x^4$, which matches.

Example 2: Differentiate $\sqrt{x}$.

Rewrite the root as a power, then apply the rule with $n = \tfrac{1}{2}$:

$$\frac{d}{dx}\left[x^{1/2}\right] = \tfrac{1}{2},x^{(1/2)-1} = \tfrac{1}{2},x^{-1/2} = \frac{1}{2\sqrt{x}}$$

Final answer: $\dfrac{1}{2\sqrt{x}}$. Reverse check: this is positive for all $x>0$, matching the fact that $\sqrt{x}$ is always increasing but flattening.

Example 3: Differentiate $\dfrac{1}{x^2}$.

Rewrite as a negative power, then apply the rule with $n = -2$:

$$\frac{d}{dx}\left[x^{-2}\right] = -2,x^{-2-1} = -2x^{-3} = -\frac{2}{x^3}$$

Final answer: $-\dfrac{2}{x^3}$. Reverse check: $\dfrac{1}{x^2}$ decreases for $x>0$, so a negative derivative is expected, and it is.

Example 4: Differentiate the polynomial $3x^4 - 5x^2 + 7x - 9$.

Combine the Power Rule with the constant-multiple and sum rules, working term by term:

$$\frac{d}{dx}\left[3x^4 - 5x^2 + 7x - 9\right]$$

$$= 3(4x^3) - 5(2x) + 7(1) - 0$$

$$= 12x^3 - 10x + 7$$

Final answer: $12x^3 - 10x + 7$. The constant $-9$ differentiates to $0$, and the degree of the polynomial drops from $4$ to $3$, as differentiation always lowers a polynomial's degree by one.

Why Does The Power Rule Work?

The algebra is short, but the reason behind it is worth seeing, because it explains why one term survives and the rest disappear.

• A derivative measures the leading response to a tiny nudge. When $x$ moves by a small amount $h$, $x^n$ changes. The binomial expansion of $(x+h)^n$ sorts that change by powers of $h$: a piece proportional to $h$, a smaller piece proportional to $h^2$, and so on.

• Only the first-order piece counts. Dividing by $h$ and letting $h \to 0$ keeps the term that was proportional to $h^1$ and crushes everything of order $h^2$ or higher. The coefficient of that first-order term is $n,x^{n-1}$.

• Geometrically, that is the tangent slope. The higher powers of $h$ describe how the curve bends; the derivative ignores the bending and reports only the straight-line slope at the point. That slope is what the tangent line in the graph above shows.

• The exponent drop is a dimension count. Differentiating turns "how much" into "how fast it changes," and that shift lowers the power by exactly one every time, the same reason position ($t^2$-type growth) differentiates into velocity ($t$-type growth).

The Power Rule is not a memorized trick, then. It is the statement that, near any point, a power curve looks like a straight line whose slope is $n,x^{n-1}$.

Who Discovered The Power Rule?

The Power Rule sits at the very start of calculus, and it arrived tangled in one of the most famous disputes in the history of mathematics.

One earlier figure paved the way for the fractional and negative cases:

• John Wallis (1616–1703, England) worked out the area under $y = x^n$ for fractional and negative $n$ in his Arithmetica Infinitorum (1656), the integral cousin of the Power Rule. His results directly influenced the young Newton, who generalized them into the binomial series that makes the rule work for every real exponent.

Where Is The Power Rule Used In The Real World?

Because so many quantities grow like powers of a variable, the Power Rule is the single most-used differentiation rule in applied mathematics.

• Physics and motion: the position of a falling object grows like $t^2$; differentiating with the Power Rule gives velocity ($\propto t$), and differentiating again gives constant acceleration. This is how the falling ball in the hook is analyzed.

• Engineering: beam-deflection, power-loss, and stress formulas are polynomials in length or current, and their rates of change (for optimization and safety margins) come straight from the Power Rule.

• Economics: cost and revenue are often modeled as power functions of quantity, and the marginal cost or revenue, the derivative, is found term by term with this rule.

• Biology and chemistry: allometric scaling laws (metabolic rate versus body mass, $\propto M^{3/4}$) and reaction-rate expressions are power functions whose derivatives use the fractional-exponent form.

• Computer graphics and machine learning: gradient-based methods differentiate polynomial and power-law loss terms; the Power Rule is what a computer applies to each such term.

One rule, "bring the power down and subtract one," runs the mathematics of motion, money, growth, and learning machines.

What Are The Most Common Mistakes With The Power Rule?

These errors are the ones documented across math-help forums and precalculus error guides, verified against reader questions about $a^x$, roots, and the "subtract one" step.

Applying the Power Rule to an exponential $a^x$.

Where it slips in:

A student sees $2^x$ or $e^x$ and writes $x\cdot 2^{x-1}$, treating a fixed base with a variable exponent as if it were a power function.

Don't do this:

Do not use $n,x^{n-1}$ when the exponent is the variable. The Power Rule is only for $x^n$ (variable base, fixed power).

The correct way:

For an exponential, use its own rule: $\frac{d}{dx}\left[a^x\right] = a^x \ln a$, so $\frac{d}{dx}\left[2^x\right] = 2^x \ln 2$ and $\frac{d}{dx}\left[e^x\right] = e^x$. Check which is the variable, the base or the exponent, before choosing a rule.

Forgetting to subtract one from the exponent.

Where it slips in:

A student brings the exponent down as a multiplier but leaves the power unchanged, writing $\frac{d}{dx}\left[x^4\right] = 4x^4$ instead of $4x^3$.

Don't do this:

Do not stop after multiplying. The rule has two steps, and the second one, dropping the exponent by one, is the one most often skipped.

The correct way:

Always complete both steps: multiply by $n$, then replace $n$ with $n-1$. So $\frac{d}{dx}\left[x^4\right] = 4x^{4-1} = 4x^3$.

Mishandling roots and negative powers.

Where it slips in:

Facing $\sqrt{x}$ or $\dfrac{1}{x^2}$, a student tries to differentiate without first rewriting them as powers, or subtracts one incorrectly from a fraction or negative exponent.

Don't do this:

Do not leave a root or a reciprocal in its original form, and do not mis-subtract. $\tfrac{1}{2}-1$ is $-\tfrac{1}{2}$, and $-2-1$ is $-3$, not $-1$.

The correct way:

Rewrite first: $\sqrt{x} = x^{1/2}$ and $\dfrac{1}{x^2} = x^{-2}$. Then apply the rule carefully: $\frac{d}{dx}\left[x^{1/2}\right] = \tfrac{1}{2}x^{-1/2}$ and $\frac{d}{dx}\left[x^{-2}\right] = -2x^{-3}$.

Practice Problems On The Power Rule

Differentiate each function using the Power Rule (with the constant-multiple and sum rules where needed). Answers follow each line.

1. $\frac{d}{dx}\left[x^8\right]$.
   (Answer: $8x^7$.)

2. $\frac{d}{dx}\left[6x^3\right]$.
   (Answer: $18x^2$.)

3. $\frac{d}{dx}\left[\sqrt[3]{x}\right]$, i.e. $x^{1/3}$.
   (Answer: $\tfrac{1}{3}x^{-2/3} = \dfrac{1}{3\sqrt[3]{x^2}}$.)

4. $\frac{d}{dx}\left[\dfrac{1}{x^3}\right]$.
   (Answer: $-3x^{-4} = -\dfrac{3}{x^4}$.)

5. $\frac{d}{dx}\left[2x^5 - 4x^2 + 9\right]$.
   (Answer: $10x^4 - 8x$.)

6. $\frac{d}{dx}\left[x^{3/2}\right]$.
   (Answer: $\tfrac{3}{2}x^{1/2} = \tfrac{3}{2}\sqrt{x}$.)

Where Should You Go Next After The Power Rule?

The Power Rule is the first differentiation rule, and several natural doors open from here.

1. Derivative formula. See the limit definition the Power Rule was built from, and the other core derivative formulas.

2. Differentiation of trigonometric functions. The next family of rules, for $\sin x$, $\cos x$, and the rest, which the Power Rule cannot reach.

3. Integration. Reverse the Power Rule to build the integral (reverse) power rule and start measuring area under curves.

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