Calculus Power Rule Formula and Examples

The calculus power rule says that the derivative of x raised to a constant power n is n times x raised to n minus 1: d(xⁿ)/dx = n xⁿ⁻¹. Enter the exponent and the x-value to calculate the derivative at that point; the calculator also shows the derivative function.

Formula

For a constant exponent n, the power rule is:

d(xⁿ)/dx = n xⁿ⁻¹

In the calculator, n is the exponent (dimensionless), x is the input value (dimensionless here), and the first result is the derivative’s value at x. The second result is the coefficient n in the derivative function n xⁿ⁻¹; it is provided as a helper, not as a complete expression for the derivative.

For the more general function f(x) = a xⁿ, where a is a constant, the rule is f′(x) = a n xⁿ⁻¹. The calculator handles the simple case a = 1. If x represents a quantity with units, the derivative’s units are the units of f divided by the units of x.

Worked Example

Using the defaults, n = 3 and x = 2:

1. Start with f(x) = x³.
2. Multiply by the exponent and reduce the exponent by one: f′(x) = 3x².
3. Substitute x = 2: f′(2) = 3 × 2² = 3 × 4 = 12.

The calculator returns a derivative value of 12, to two decimal places. The derivative function is 3x²; its coefficient is 3.

What the Power Rule Means

A derivative measures how quickly a function changes as its input changes. Geometrically, it gives the slope of the function’s tangent line at a point. For f(x) = x³, the slope depends on x: the derivative function is 3x², so at x = 2 the slope is 12. This is not the original function’s value, which is 2³ = 8.

The rule is useful because it turns a common family of functions into a direct calculation. Instead of using the limit definition of a derivative every time, you can apply the exponent rule whenever a term is a constant times a power of x.

How to Apply It to Expressions

Apply the rule to each power term separately, and keep constant factors in place. For example:

• f(x) = 5x⁴ gives f′(x) = 20x³.
• g(x) = 3x² − 7x + 4 gives g′(x) = 6x − 7. The constant 4 has derivative zero.
• h(x) = x⁻² gives h′(x) = −2x⁻³, equivalently −2/x³.
• p(x) = √x can be written x¹ᐟ², so p′(x) = ½x⁻¹ᐟ² = 1/(2√x), for x > 0.

For a polynomial, a practical sequence is to rewrite terms as powers where useful, differentiate each term, and simplify. The power rule does not replace other rules when functions are combined in more complex ways: products, quotients, and compositions may require the product, quotient, or chain rule.

Special Exponents and Domain Limits

The rule also covers zero and negative integer exponents, with the usual restrictions on where the original function is defined. When n = 0, x⁰ is 1 (for the ordinary nonzero-domain interpretation), and its derivative is 0. When n = 1, the derivative of x is 1. For n = −1, the function is 1/x and its derivative is −1/x², defined only when x is not zero.

Fractional powers need care around the real-number domain. For instance, √x is real for x ≥ 0, but its derivative formula 1/(2√x) is not finite at x = 0. More generally, a function may be defined at a point where its derivative is not. The power-rule expression should therefore be used only where the original function and the derivative are defined.

Common Mistakes and Limits

• Forgetting to lower the exponent: xⁿ becomes n xⁿ⁻¹, not n xⁿ.
• Dropping a coefficient: the derivative of 5x⁴ is 5 × 4x³, or 20x³.
• Confusing a value with a function: 3x² is the derivative function; at x = 2 its value is 12.
• Misreading negative powers: x⁻² means 1/x², and it is undefined at x = 0.
• Using the rule on the wrong structure: the derivative of (x + 1)³ requires the chain rule as well as the power rule, giving 3(x + 1)².

The calculator accepts any numerical exponent and evaluates the stated expression at the supplied x-value. That arithmetic result does not establish that a real-valued function or derivative exists there. In particular, negative bases with noninteger exponents can be outside the real-number domain, and a denominator or negative power can make an expression undefined at zero. Check the function’s domain before interpreting a numerical output.

Frequently Asked Questions

What is the power rule in calculus?
The derivative of xⁿ, for a constant exponent n, is n xⁿ⁻¹. Multiply by the old exponent, then subtract one from that exponent.

What is the derivative of xⁿ?
It is n xⁿ⁻¹. For example, the derivative of x⁵ is 5x⁴.

What is the power rule for a negative exponent?
It works the same way: x⁻³ differentiates to −3x⁻⁴, wherever the original function is defined. Since x⁻³ is undefined at zero, the derivative is also not defined there.

What is the derivative of a constant?
Zero. A constant does not change as x changes, so its rate of change is zero.

Does the power rule work for square roots?
Yes. Rewrite √x as x¹ᐟ² and apply the rule to get 1/(2√x). This derivative formula applies for x > 0.

Can I use the power rule for (2x + 1)³?
Not by itself: the expression is a power of an inner function. Use the chain rule too, giving 3(2x + 1)² × 2 = 6(2x + 1)².

Related

• Angle Calculator: Elevation, Distance and Height
• Angle of Elevation Calculator: Find Horizontal Distance
• Binary Floating Point to Decimal Calculator in Java

in n | Exponent | - | 3 | - | -
in x | x value | - | 2 | - | -
out derivative | Derivative at x | - | n * x^(n - 1) | 2
out function | Derivative function coefficient | - | n | 0