AP® Calculus AB · Unit 2 · Topic 2.5
Applying the Power Rule
The power rule says the derivative of xn is n xn-1: slide the exponent down front, then subtract one from it. It turns slow limit calculations into a quick mechanical step for any polynomial term.
The power rule exists because of what happens when you expand (x+h)n inside the limit definition of the derivative. The limit definition says f'(x) = lim as h → 0 of [f(x+h) - f(x)] / h. For f(x) = xn, expanding (x+h)n with the binomial theorem gives xn + n xn-1 h + (terms with h2 or higher) + hn. Subtract xn and divide by h, and every surviving term still has at least one factor of h except one: n xn-1. As h → 0, every term with leftover h vanishes, and n xn-1 is all that remains. That single surviving term is the whole rule: bring the exponent down as a coefficient, then lower the exponent by one.
Once you trust where it comes from, applying it is mechanical. Differentiate term by term: for a xn, the derivative is a n xn-1. Constants disappear entirely, because a constant is really c x0, a flat horizontal line with slope zero everywhere, so differentiating it correctly gives 0, not the constant itself. The common slip is treating a constant like it should survive differentiation unchanged — it doesn't, because it never had an x to begin with, and the rate of change of a flat value is always zero.
This rule is the workhorse of Unit 2 because almost every function you differentiate next — sums, products, quotients, chains — eventually breaks down into power-rule pieces. The one place it changes shape is when the exponent itself is a variable, as in bx; that case needs logarithmic differentiation, not the power rule, since here the base is fixed and the exponent is what varies.
Key terms
- Power rule
- The rule stating that the derivative of xn is n xn-1.
- Binomial expansion
- The algebraic expansion of (x+h)n into a sum of terms with decreasing powers of x and increasing powers of h.
- Limit definition of the derivative
- The formula f'(x) = lim as h → 0 of [f(x+h) - f(x)] / h that defines the derivative from first principles.
- Constant term
- A term with no variable factor, equivalent to x0, whose derivative is always 0.
Watch it explained
- How it worksHow does the binomial expansion inside the limit definition produce the power rule?
- Worked exampleFind the derivative of f(x) = 7x5 - 2x3 + 9 using the power rule.
- Common mistakeWhy isn't the derivative of a constant term the constant itself?
- Going furtherWhat happens to the power rule when the exponent is a variable instead of a number?
Check yourself
End of topic.