The formal definition of a derivative — the difference quotient — can take pages of algebra for a single function. The power rule collapses all of that into one line. It is the most-used differentiation shortcut in calculus, showing up in everything from polynomial curves to velocity problems to marginal cost in economics.
In this tutorial you will master the universal power rule formula, handle negative and fractional exponents with confidence, and differentiate full polynomials term by term.
What Is the Power Rule for Derivatives?
For any variable raised to a constant real-number exponent:
d/dx[x^n] = n · x^(n-1)
In plain terms: bring the exponent down to the front, then subtract 1 from the exponent. If the term has a constant multiplier, it rides along untouched:
d/dx[c · x^n] = c · n · x^(n-1)
If you want to review how linear rates connect to slope, see our breakdown on slope-intercept form explained. For practical fractional manipulation when adjusting exponents, explore our tutorial on multiplying and dividing fractions keep change flip. You can also verify derivative expansions using the CalcSolver calculation tool.
The 3-Step Power Rule Framework
- Rewrite into power form: convert radicals into fractional exponents (√x = x^(1/2)) and move denominator variables upstairs with negative exponents (1/x² = x⁻²).
- Bring the power forward: multiply the coefficient by the current exponent.
- Subtract 1 from the power: reduce the exponent by exactly 1 to finish the term.
Worked Example 1: Standard Polynomial
Find the derivative of:
f(x) = 4x^3 – 5x^2 + 7x – 9
Apply the power rule to each term individually:
- 4x³: bring down 3 and subtract 1: 4 · 3 · x³⁻¹ = 12x²
- −5x²: bring down 2 and subtract 1: −5 · 2 · x²⁻¹ = −10x
- 7x: the exponent is 1: 7 · 1 · x¹⁻¹ = 7 · x⁰ = 7 (anything to the 0 power is 1)
- −9: a standalone constant never changes, so its derivative is 0
Final derivative: f'(x) = 12x² − 10x + 7.
Worked Example 2: Negative Exponents (Variables in the Denominator)
Differentiate:
f(x) = 3/x^2
Step 1: Rewrite with a negative exponent
Move x² upstairs by flipping the exponent’s sign:
f(x) = 3x^(-2)
Step 2: Apply the power rule
3 · (-2) · x^(-2-1)
Step 3: Simplify the exponent
Subtracting 1 from a negative number moves it further left on the number line: −2 − 1 = −3.
f'(x) = -6x^(-3)
Step 4: Rewrite without negative exponents
f'(x) = -6/x^3
Worked Example 3: Fractional Exponents (Square Roots)
Differentiate:
f(x) = 5√(x)
Step 1: Convert the radical to a rational exponent
A square root is a power of 1/2:
f(x) = 5x^(1/2)
Step 2: Multiply by the exponent
5 · 1/2 · x^((1/2) – 1)
Step 3: Compute the new fractional power
(1/2) − 1 = (1/2) − (2/2) = −1/2
f'(x) = 5/2 · x^(-1/2)
Step 4: Format in clean radical notation
Move the negative fractional power back underneath as a root:
f'(x) = 5/2√(x)
Worked Example 4: A Cube Root Squared
Differentiate f(x) = ∛(x^2), i.e. f(x) = x^(2/3).
- Bring down the exponent: (2/3) · x^((2/3) − 1)
- Subtract: (2/3) − 1 = (2/3) − (3/3) = −1/3
- Rewrite as a radical: f'(x) = 2/3∛(x)
Radical rule of thumb: never try to differentiate an expression while it’s trapped under a root symbol. Convert to a fractional exponent first, differentiate, then convert back. This one habit prevents the majority of power-rule errors.
3 Common Mistakes to Avoid
- Adding 1 instead of subtracting: differentiation always decreases the exponent by 1. Adding 1 belongs to integral calculus (antiderivatives).
- Subtracting negative numbers incorrectly: differentiating x⁻⁴ gives x⁻⁵, not x⁻³. Moving one step down the number line from −4 lands on −5.
- Dropping the coefficient: in a term like 5x^(1/2), the 5 stays multiplied the whole time — it doesn’t get absorbed into the exponent.
Rewrite, bring down, subtract one: with that rhythm you can differentiate polynomials, rational terms, and radicals quickly and accurately — no difference quotient required.