The power rule is a method to find the derivative of a variable raised to an exponent, and gives us a quick way to differentiate—that is, to take the derivative of—functions like x^2 and x^3. The power rule is used frequently since functions like that are ubiquitous throughout calculus.
The power rule is a calculus shortcut used to find the derivative of a variable raised to a power. The core formula is
How to Use the Rule
- The Basic Formula: For f(x) = x⁴, you bring down the 4 and subtract 1 from the power, resulting in 4x³.
- With a Coefficient: If there is a constant in front of x, multiply the exponent by that constant. For example, the derivative of 5x³ is 5 ⋅ 3x² = 15x².
- The Constant Rule: The derivative of any standalone number (constant) is 0.
Key Concepts at a Glance
| Concept | Statement | AP© Standard |
|---|---|---|
| Power Rule | If(x) = xr, then f'(x) = r · xr-1 | CED Topic 2.1 — Defining Average and Instantaneous Rate |
| Applies to | All real exponents: positive, negative, and fractional | Essential for FRQ Part A polynomial derivatives |
| Fractional exponents | √x = x½ derivative ½x-½ | Rewrite radicals before differentiating |
| Negative exponents | 1/x2 = x-2 derivative -2x-3 | Rewrite fractions before differentiating |
| Constant rule | d/dx[c] = O for any constant c | Constants vanish; don’t forget leading coefficients |
Why Use the Power Rule?
Once you understand what a derivative represents, you start seeing why it’s useful everywhere. [1]
Physics: If you know the position of an object as a function of time — say, 𝑠(𝑡) =5𝑡2 for a falling object — then the derivative 𝑠′(𝑡) =10𝑡 gives you the velocity at any moment 𝑡. The derivative of velocity gives you acceleration. This is the mathematical foundation of classical mechanics.
Finding maximums and minimums: Imagine you’re running a business and your profit is modeled by 𝑃(𝑥) =−𝑥2 +100𝑥, where 𝑥 is the number of units you produce. You want to know how many units to produce to maximize profit. Where is profit highest? At the peak of that parabola — which is exactly where the slope is zero. Set the derivative equal to zero and solve. This is one of the most powerful applications of calculus.
The “highest point” of anything: A ball thrown upward follows a curved path. At the very peak, it momentarily has zero vertical velocity — it stops going up and hasn’t yet started coming down. At that exact moment, the slope of the height-vs-time graph is zero. The derivative equals zero. Calculus lets you find that moment precisely.
Engineering, economics, biology — derivatives are everywhere a rate of change matters. And rates of change matter almost everywhere.
Power Rule Proof
Case 1: Variable
Let us begin with the simple algebraic expression y = x2. Remember that the fundamental notion about calculus is the idea of growing. Now as y and x2 are equal, it is clear that if y grows, x2 will also grow. And if x2 grows, then y will also grow. What we need to find is the proportion between the growing of y and the growing of x. In other words, our task is to find out the ratio between dy and dx, or, in brief, to find the value of dy/dx. [2]
y = x2 [Equation 1-1]
Let x grow a little bit bigger and become x + dx; similarly, y will grow a bit bigger and will become y + dy.
y + dy = (x + dx)2 [Equation 1-2]
Expanding the right side of the equation, we get:
y + dy = x2 + 2x*dx +(dx)^2 [Equation 1-3]
Since (dx)^2 is negligible, we can discard it as inconsiderable in comparison with the other terms. (Think of dx as equal to 10-10 or smaller, therefore (dx)2 = 10-20.) Our equation then becomes:
y + dy = x2 + 2x*dx [Equation 1-4]
Now substitute y = x2 into Equation 1-4 and simplify the equation. We then get:
x2 + dy = x2 + 2x*dx [Equation 1-5a]
dy = 2x * dx [Equation 1-5b]
Dividing both sides of the equation by dx , we get
dy/dx = 2x (QED)
Case 2: Constant
Let’s take a similar approach as we did in Case 1.
y = x3 + 5 [Equation 2-1]
Then:
y + dy = (x + dx)3 + 5 = x3 + 3x2 * dx + 3x(dx)2 + (dx)3 + 5 [Equation 2-2]
Neglecting the small quantities of higher orders, i.e., 3x(dx)2 + (dx)3, this becomes
y + dy = x3 + 3x2 * dx + 5 [Equation 2-3]
Substituting the original y = x3 + 5 into the above equation and simplifying, we get
x3 + 5 + dy = x3 + 3x2 * dx + 5 [Equation 2-4a]
dy/dx = 3x2 [Equation 2-4b]
So, the 5 disappears as it adds nothing to the growth of x, and does not enter into the differential coefficient. [2]
References
[1] Free Math Help. “Derivative of a Polynomial.” Accessed June 10, 2026. https://www.freemathhelp.com/derivative-of-polynomial/.
[2] Thompson, Silvanus P. 1914. Calculus Made Easy. 2nd ed. The Macmillan Company. https://www.gutenberg.org/files/33283/33283-pdf.pdf.
Additional Reading
“2.5 Applying the Power Rule.” Calculus. Accessed June 11, 2026. https://calculus.flippedmath.com/25-applying-the-power-rule.html.
CUEMATH. “Power Rule – Formula, Proof, Applications.” Accessed June 11, 2026. https://www.cuemath.com/calculus/power-rule/.
CUEMATH. “Power Of a Power Rule – Formula, Examples.” Accessed June 11, 2026. https://www.cuemath.com/algebra/power-of-a-power-rule/.
GeeksforGeeks. “Applications of Power Rule.” GeeksforGeeks, June 15, 2021. https://www.geeksforgeeks.org/maths/applications-of-power-rule/.
GeeksforGeeks. “Power Rule.” GeeksforGeeks, October 27, 2020. https://www.geeksforgeeks.org/maths/power-rule/.
Guichard, David. “3.1: The Power Rule.” Libretexts, November 7, 2013. https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/03%3A_Rules_for_Finding_Derivatives/3.01%3A_The_Power_Rule.
[ ℰ ] Pierce, Rod. “Derivative Rules.” Math Is Fun. Accessed June 10, 2026. https://www.mathsisfun.com/calculus/derivatives-rules.html.
The Albert Team. “Power Rule for Derivatives: AP® Calculus AB-BC Review.” Albert Blog & Resources, May 12, 2025. https://www.albert.io/blog/power-rule-for-derivatives-ap-calculus-ab-bc-review/.
[ ℰ ] Zemke, Drew. “What Is the Power Rule?” Outlier, October 6, 2021. https://articles.outlier.org/what-is-the-power-rule.
Videos
Notes
We call the ratio dy/dx “the differential coefficient of y with respect to x“. A differential coefficient is the older term for a mathematical derivative. It measures the instantaneous rate of change of a dependent variable with respect to an independent variable. Geometrically, it represents the slope of a curve at any given point. It is most commonly written as dy/dx or f'(x).
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