Power Function Calculator

Evaluate a power function f(x) = a·xb at a chosen x, then find its derivative and antiderivative using the power rule.

Quick Facts

Power function
f(x) = a·x^b
a is a nonzero coefficient; b is a fixed real exponent; x is the base.
Power rule (derivative)
d/dx[a·x^b] = a·b·x^(b-1)
Multiply by the exponent, then reduce the exponent by 1.
Power rule (integral)
∫a·x^b dx = a/(b+1)·x^(b+1) + C
Valid for b ≠ -1; when b = -1 the antiderivative is a·ln|x| + C.
Domain note
x > 0 when b is not an integer
Fractional powers of negative numbers are generally not real.

Your Results

Calculated
f(x)
-
f(x) = a·x^b evaluated at x
f'(x)
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Derivative value at x, via the power rule
Derivative formula
-
f'(x) = a·b·x^(b-1)
Antiderivative formula
-
∫f(x) dx, general form

Ready

Enter a, b, and x, then press Calculate.

Formula and Method for the Power Function Calculator

A power function has the form f(x) = a·xb, where a is a nonzero coefficient, b is a fixed real-number exponent, and x is the variable base. This is the defining trait that separates a power function from an exponential function like f(x) = a·bx — in a power function the variable is the base being raised to a fixed power; in an exponential function the variable sits in the exponent instead. This calculator evaluates f(x) at a chosen x, then applies the power rule to report the derivative and the general antiderivative.

Evaluating, differentiating, and integrating

Given a, b, and x, the calculator first computes the function value f(x) = a·xb directly. It then applies the power rule for differentiation: the derivative of a·xb is f′(x) = a·b·x(b−1) — multiply by the exponent, then subtract 1 from the exponent. Finally it applies the power rule for integration: the general antiderivative is ∫a·xb dx = [a/(b+1)]·x(b+1) + C, valid whenever b ≠ −1. The one exception is b = −1 (i.e., f(x) = a/x), where dividing by (b+1) = 0 is undefined; in that special case the antiderivative is instead a·ln|x| + C.

Domain restrictions to watch for

  • Fractional exponents need x > 0: when b is not a whole number, raising a negative x to that power generally does not produce a real number (for example, (−4)0.5 is not real), so this calculator requires x > 0 whenever b is not an integer.
  • Negative exponents need x ≠ 0: if b < 0, then x = 0 makes xb undefined (division by zero), and the same is true for the derivative whenever (b − 1) < 0.
  • Integer exponents work for any nonzero x: whole-number exponents (positive or negative) are defined for negative x as well — for example, (−2)3 = −8 is perfectly valid.

Real-world applications

  • Physics and engineering use power functions for relationships like area scaling with the square of a length (b = 2) or gravitational force scaling with the inverse square of distance (b = −2).
  • Biology uses power-law (allometric) scaling to relate body mass to metabolic rate or other traits.
  • Calculus courses use simple power functions like x², x³, and √x = x0.5 as the standard first examples for practicing the power rule for derivatives and integrals.

Frequently Asked Questions

What is a power function?
A power function has the form f(x) = a·x^b, where a is a nonzero coefficient and b is a fixed real exponent, and the variable x is the base. This is different from an exponential function, such as f(x) = a·b^x, where the variable is in the exponent instead of the base.
How do you find the derivative of a power function?
Use the power rule: the derivative of f(x) = a·x^b is f'(x) = a·b·x^(b-1). For example, the derivative of f(x) = 2x^3 is f'(x) = 6x^2.
How do you integrate a power function?
For b ≠ -1, the antiderivative is ∫a·x^b dx = [a/(b+1)]·x^(b+1) + C. When b = -1 (i.e., f(x) = a/x), the power rule for integration does not apply and the antiderivative is instead a·ln|x| + C.
Why does x need to be positive for some exponents?
When the exponent b is not a whole number, raising a negative base to that power generally does not produce a real number (for example, the square root of a negative number is not real). This calculator therefore requires x > 0 whenever b is not an integer; for whole-number exponents, any nonzero x is valid.

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