Difficulty: Beginner | Prerequisites: Understanding of exponents and basic algebra
Tags: power rule, derivative, differentiation, exponent, calculus I, brief calculus, f'(x), dy/dx, negative exponent, fractional exponent, ASU
The power rule is the single most-used shortcut in differential calculus. Before it, finding a derivative meant setting up the limit definition every time, which is slow and error-prone for anything beyond the simplest polynomials. Once you have the power rule, you can differentiate any term of the form xⁿ in seconds, and it extends naturally to negative and fractional exponents, meaning roots and reciprocals are covered too.
You should already be comfortable rewriting expressions like 1/x⁴ as x⁻⁴ and ⁴√x as x^(1/4). If that conversion feels shaky, revise your exponent rules before going further.
If f(x) = xⁿ, its derivative is f'(x) = n · xⁿ⁻¹. Multiply by the current exponent, then reduce the exponent by one. This works for positive, negative, and fractional values of n, so it handles polynomials, reciprocals, and roots alike.
Derivative
The instantaneous rate of change of a function at a given point. In simple terms, it tells you how fast the output is changing as the input changes.
Power rule
If f(x) = xⁿ where n is any real constant, then f'(x) = n · xⁿ⁻¹. Think of it as "bring the exponent down as a multiplier, then knock the exponent down by one."
Differential notation
The different ways of writing "the derivative of f with respect to x." Common forms: f'(x), dy/dx, and d/dx [f(x)]. All three mean the same thing.
Negative exponent
A way of writing a reciprocal as a power: x⁻ⁿ = 1/xⁿ. This lets you apply the power rule to functions with variables in the denominator.
Fractional exponent
A way of writing a root as a power: x^(1/n) = ⁿ√x. This lets you apply the power rule to root functions.
If n is any real constant and f(x) = xⁿ, then f'(x) = n · xⁿ⁻¹.
The rule works the same way regardless of whether n is a positive integer, a negative integer, or a fraction.
Example: f(x) = 6x⁵
Bring down the exponent: 6 · 5 = 30
Reduce the exponent by one: 5 − 1 = 4
Result: f'(x) = 30x⁴
When the variable sits in a denominator, rewrite it with a negative exponent first, then apply the rule as normal.
Example: h(t) = 1/t⁴
Rewrite: h(t) = t⁻⁴, so n = −4
Differentiate: h'(t) = −4 · t⁻⁴⁻¹ = −4t⁻⁵
Back to positive exponent form: h'(t) = −4/t⁵
Roots become fractional powers. Apply the rule, then convert back if the question asks for radical form.
Example: s(x) = 1/⁴√x
Rewrite: s(x) = x⁻¹ᐟ⁴, so n = −1/4
Differentiate: s'(x) = (−1/4) · x^(−1/4 − 1) = (−1/4) · x⁻⁵ᐟ⁴
In radical form: s'(x) = −1 / (4 · ⁴√x⁵)
The power rule handles decimal exponents just as well.
Example: f(x) = 10x⁶·⁸ + 25x⁰·²
Differentiate term by term: f'(x) = 10 · 6.8 · x⁵·⁸ + 25 · 0.2 · x⁻⁰·⁸
Simplify: f'(x) = 68x⁵·⁸ + 5x⁻⁰·⁸
When a function is given as a fraction or a product, simplify or distribute first so every term is in the form axⁿ.
Example: g(x) = x²[x³ + 2/x]
Distribute x²: g(x) = x⁵ + 2x
Differentiate: g'(x) = 5x⁴ + 2
Example: h(x) = (8x⁴ + 18x²) / (2x⁴)
Split and simplify: h(x) = 4 + 9x⁻¹
The constant 4 vanishes: h'(x) = −9x⁻²
Rewrite: h'(x) = −9/x²
\frac{d}{dx}[x^n] = n \cdot x^{n-1}\frac{d}{dx}[c] = 0 \quad \text{(any constant)}Exponent conversion reminders:
\frac{1}{x^n} = x^{-n} \qquad \sqrt[n]{x} = x^{1/n}The power rule is how you find velocity from a position function (position is often a polynomial in t, and its derivative is speed). In economics, marginal cost and marginal revenue are derivatives of cost and revenue functions, which are frequently power functions. Any time a model uses a polynomial to approximate real data, the power rule is the tool for analysing how that quantity changes.
Students often forget to rewrite roots and reciprocals as powers before differentiating. The power rule only works on the form xⁿ, so 1/x⁴ must become x⁻⁴ first.
A common error is subtracting the exponent incorrectly: the new exponent is n − 1, not n − n. For x⁵ the derivative is 5x⁴, not 5x⁰.
Students sometimes try to apply the power rule to constants. The derivative of any standalone constant (like 7 or ³√9) is zero, not one.
When the exponent is negative, students sometimes drop the negative sign during the multiplication step. If n = −4, then n · xⁿ⁻¹ = −4x⁻⁵, not 4x⁻⁵.
⚠️ Expect at least one problem where you must rewrite a root or reciprocal before differentiating. The rewrite is often the part that earns partial credit.
⚠️ Questions that say "simplify your answer" or "write with positive exponents" mean you need to convert negative exponents back to fraction form at the end.
⚠️ The derivative of a constant is zero. If a term has no variable in it (like ³√9), it disappears on differentiation. This gets tested regularly.
True or false: The derivative of x⁷ is 7x⁷.
Fill in the blank: To differentiate 1/x³, first rewrite it as x____.
True or false: The derivative of the constant 12 is 0.
Fill in the blank: If f(x) = x^(1/2), then f'(x) = ____ · x^(____).
True or false: The power rule only works when the exponent is a positive integer.
Answers: 1. False (7x⁶). 2. x⁻³. 3. True. 4. (1/2) · x^(−1/2). 5. False (works for any real n).
Q: Find the derivative of f(x) = 4x⁹.
A: f'(x) = 36x⁸
Q: Differentiate g(t) = 1/t⁶. Write your answer with positive exponents.
A: Rewrite as t⁻⁶. Then g'(t) = −6t⁻⁷ = −6/t⁷.
Q: Find the derivative of y = ³√x.
A: Rewrite as x^(1/3). Then y' = (1/3)x^(−2/3) = 1/(3·³√x²).
Q: If f(x) = 5x⁴ − 7, what is f'(x)?
A: f'(x) = 20x³. The constant −7 drops away.
Q: Differentiate h(x) = x²(x³ + 2/x). Simplify first.
A: Distribute to get x⁵ + 2x. Then h'(x) = 5x⁴ + 2.
This connects directly to the sum and difference rules for derivatives, which let you differentiate polynomials term by term. It also feeds into tangent-line problems, where you use the derivative to find the slope at a specific point.
Later in the course, the power rule becomes a building block inside the chain rule, product rule, and quotient rule, all of which assume you can already differentiate individual power terms quickly.
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