The power rule and where it comes from

TL;DR:

• The power rule says the derivative of xⁿ is n·xⁿ⁻¹.

• It comes from expanding (x + h)ⁿ and noticing only the term with one h survives as h shrinks to zero.

• Constants drop out, constant multiples stay, and sums are differentiated term by term.

Finding a derivative from a table of shrinking intervals works, but it is slow. A rule is faster, and the power rule covers every polynomial. It is not magic: it falls out of a short piece of algebra that you can do yourself for x³.

The limit definition

The derivative of f at x is the value the average rate (f(x + h) − f(x)) / h approaches as h shrinks toward 0. The top is the change in output, and h is the change in input. We cannot just set h = 0 at the start, because that would give 0/0. We simplify first, so the h in the bottom cancels, and only then let h go to zero.

Worked example: the derivative of x³ from the limit definition

1. Write the difference quotient: ((x + h)³ − x³) / h.

2. Expand (x + h)³ = x³ + 3x²h + 3xh² + h³.

3. Subtract x³: the top becomes 3x²h + 3xh² + h³.

4. Divide each term by h: 3x² + 3xh + h².

5. Let h → 0. The terms with h vanish, leaving 3x².

So the derivative of x³ is 3x². Check with a number. At x = 2, the rule gives 3 × 4 = 12. Now take a small h: (2.001³ − 8) / 0.001 = (8.012006 − 8) / 0.001 = 12.006. It is close to 12, and moves closer as h shrinks.

Why the pattern works for any power

Look at which term survives. In (x + h)³, the first term is x³ and cancels with the −x³. The next term, 3x²h, has exactly one h, so after dividing by h it becomes 3x². Every other term has at least h², so one h is still left after division and the term vanishes as h goes to zero. The surviving coefficient, 3, is the number of ways to choose which of the three brackets supplies the h.

The same reasoning for (x + h)ⁿ gives n ways to choose the bracket, and the leftover is xⁿ⁻¹. That is the power rule: the derivative of xⁿ is n·xⁿ⁻¹. Check it for n = 2: expand (x + h)² = x² + 2xh + h², subtract x², divide by h, get 2x + h, and the limit is 2x. The rule gives 2x¹ = 2x. It works.

Using the rule on whole expressions

• Constant multiples stay. The derivative of 5x⁴ is 5 × 4x³ = 20x³.

• Sums go term by term. The derivative of 5x⁴ − 3x² + 7x + 2 is 20x³ − 6x + 7.

• A constant has derivative 0. It does not change, so its rate of change is zero.

• The derivative of x is 1. Treat x as x¹, so 1 × x⁰ = 1.

Evaluate at a point. At x = 1, the derivative 20x³ − 6x + 7 is 20 − 6 + 7 = 21. That is the slope of the tangent line to 5x⁴ − 3x² + 7x + 2 at x = 1. Check numerically: the function at 1 is 5 − 3 + 7 + 2 = 11, and at 1.001 it is about 11.021, so the slope is about 0.021 / 0.001 = 21.

Try one yourself: 4x³ − 2x² + x − 9

Apply the rule to each term: 4 × 3x² = 12x², −2 × 2x = −4x, 1, and the constant disappears. The derivative is 12x² − 4x + 1. At x = 2, that is 48 − 8 + 1 = 41. Check numerically: the function at 2 is 32 − 8 + 2 − 9 = 17, and at 2.001 it is about 17.041, so the slope is about 0.041 / 0.001 = 41.

Roots and reciprocals are powers too

Write a root as a fractional power: √x = x^(1/2). Then the derivative is (1/2)x^(−1/2) = 1 / (2√x). At x = 4 that is 1 / (2 × 2) = 0.25. Check: (√4.01 − 2) / 0.01 ≈ (2.0025 − 2) / 0.01 = 0.25. Similarly 1/x = x⁻¹ has derivative −x⁻² = −1/x². At x = 2 the slope is −1/4, negative because the curve falls.

How to check your answer

• Degree drops by one. A cubic should give a quadratic.

• Numeric check. Take a small h and compute the difference quotient, as above.

• Sign. If the function rises at that point, the derivative should be positive.

• Special case. Does the rule give 0 for a constant and 1 for x?

Common mistakes

• Forgetting to multiply by the old exponent: writing x² as the derivative of x³.

• Subtracting 1 from the wrong thing: 3x³, not 3x².

• Applying the power rule to 2ˣ. The variable is in the exponent there, and the rule does not apply.

• Differentiating a product as the product of derivatives.

• Forgetting that the derivative of a constant term is 0.

Where you will use it next

Once you have the power rule, composite expressions like (3x + 1)⁵ need one more tool, found in the chain rule with one picture. The idea of a derivative as a rate is reviewed in what a derivative measures. You can write down worked examples as you go: a sheet of solved problems is a good way to learn rules like this one.

Frequently asked questions

Does the power rule work when the exponent is negative or a fraction?

Yes. It holds for any real constant exponent, as long as x is in the domain, for example x above 0 for fractional powers.

Why can I not set h = 0 right away?

Because the quotient would be 0/0. After simplification, the h cancels and the limit can be read off.