Power Functions f(x) = xⁿ: Graphs, Derivative, Integral and Rules
Power functions f(x) = xⁿ for whole, negative and fractional n: graphs, domains, the derivative nxⁿ⁻¹, the integral and the n = -1 exception.
Algebraic functions Updated 2 min read
At a glance
- Domain
- All reals for whole n \ge 0; x \ne 0 for negative n
- Derivative
- nx^{n-1}
- Integral
- \frac{x^{n+1}}{n+1} + C
- Exception
- \int x^{-1}\,dx = \ln|x| + C
- Even or odd
- Even for even whole n, odd for odd whole n
- Passes through
- (1, 1) for every n
On this page
A power function raises the input to a fixed exponent: f(x) = x^n. The family includes parabolas, cubics, reciprocals and roots, and one rule, the power rule, differentiates every member.
Formula
The exponent n is a constant. More generally f(x) = kx^n with a constant coefficient k. Common members:
| n | Function | Name |
|---|---|---|
| 0 | 1 | constant |
| 1 | x | identity line |
| 2 | x^2 | parabola |
| 3 | x^3 | cubic |
| -1 | \frac{1}{x} | reciprocal |
| \frac{1}{2} | \sqrt{x} | square root |
Graphs of power functions
How the exponent shapes the graph:
- Even whole n (x^2, x^4, \ldots): U-shaped, even function, range [0, \infty).
- Odd whole n (x^3, x^5, \ldots): S-shaped, odd function, range all reals.
- Negative n (x^{-1}, x^{-2}, \ldots): undefined at 0, with a vertical asymptote at x = 0 and a horizontal asymptote y = 0.
- Fractional n = \frac{p}{q}: an even root (q even) restricts the domain to x \ge 0; an odd root allows all reals.
- On (0, 1) higher powers are smaller; on (1, \infty) higher powers are larger.
Derivative of a power function
Bring the exponent down, then subtract one. This holds for every real constant n wherever x^n is differentiable. Note that for 0 < n < 1, such as \sqrt{x}, the derivative blows up at x = 0, where the graph has a vertical tangent.
Integral of a power function
Add one to the exponent, then divide by the new exponent. The case n = -1 would divide by zero; instead
See the reverse power rule.
Worked examples
Example 1. \frac{d}{dx}x^7 = 7x^6.
Example 2: negative power. \frac{d}{dx}\frac{4}{x^3} = \frac{d}{dx}4x^{-3} = -12x^{-4} = -\frac{12}{x^4}.
Example 3: fractional power. \frac{d}{dx}\sqrt[3]{x} = \frac{d}{dx}x^{1/3} = \frac{1}{3}x^{-2/3}.
Example 4: integral. \int x^{3/2}\,dx = \frac{x^{5/2}}{5/2} + C = \frac{2}{5}x^{5/2} + C.
Example 5: definite integral. \int_1^4 \frac{1}{\sqrt{x}}\,dx = \left[2x^{1/2}\right]_1^4 = 4 - 2 = 2.
Common mistakes
- Using the power rule on 2^x. That is an exponential function; its derivative is 2^x\ln 2.
- Subtracting one from a negative exponent in the wrong direction. x^{-3} differentiates to -3x^{-4}, not -3x^{-2}.
- Applying the integral formula at n = -1. \frac{x^0}{0} is meaningless; use \ln|x|.
- Not rewriting first. Turn \frac{1}{x^2} into x^{-2} and \sqrt{x} into x^{1/2} before using either rule.
How it connects
Sums of power functions with whole exponents are polynomial functions, and power series extend that idea to infinitely many terms. For a deeper walkthrough with practice problems, read the guide to the power rule and to antiderivatives.
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The power rule
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