Power Rule
In short: Power rule says that x raised to a constant exponent n has derivative n times x raised to n minus 1: drop the exponent in front, then reduce it by one.
AP Calculus ABCalculusAP Calculus AB Unit 2
What is Power Rule?
The power rule is the workhorse of differentiation. It handles any term of the form \(x^n\), and \(n\) may be positive, negative or a fraction.
In symbols, \(\frac{d}{dx}\left[x^n\right] = n\,x^{n-1}\). The old exponent hops down to become a multiplier, and the new exponent is one smaller. A number sitting in front of the power just stays put and multiplies the result.
Roots and reciprocals qualify too, once you rewrite them: \(\sqrt[3]{x} = x^{1/3}\) and \(\frac{1}{x^4} = x^{-4}\).
How to spot it
- The variable is raised to a plain number, as in x^7, x^(-2) or x^(3/4).
- A cube root, square root or 1 over a power of x appears, and the question asks for a slope or rate.
- A position function like s(t) = t^(5/2) is given and the velocity is requested.
- You see a polynomial, and the power rule is the fastest route to its derivative.
The method
- Convert any root or reciprocal into an exponent form such as \(x^{1/2}\) or \(x^{-3}\).
- Multiply the coefficient by the exponent.
- Subtract 1 from the exponent, taking care with fractions: \(\frac{5}{2} – 1 = \frac{3}{2}\).
- Simplify, then substitute a number if the question asks for the value at a specific input.
Power Rule: solved example
Problem
A kite drifts so that its horizontal distance from the launch point is \(s(t) = t^{5/2} + 3t\) feet after \(t\) seconds. Find its horizontal speed at \(t = 4\).
Solution
Apply the rule to each term: \(s^{\prime}(t) = \frac{5}{2}t^{3/2} + 3\).
At \(t = 4\): \(4^{3/2} = \left(\sqrt{4}\right)^3 = 8\), so \(s^{\prime}(4) = \frac{5}{2}(8) + 3 = 20 + 3\). The kite moves at 23 feet per second.
Try it
Question 1
Differentiate \(g(x) = 9x^{-4}\).
Show the answer
Multiply 9 by \(-4\) to get \(-36\), and lower the exponent to \(-5\): \(g^{\prime}(x) = -36x^{-5} = -\frac{36}{x^5}\).
Question 2
Find \(\frac{d}{dx}\left[\sqrt[3]{x}\right]\) at \(x = 8\).
Show the answer
\(x^{1/3}\) has derivative \(\frac{1}{3}x^{-2/3}\). Since \(8^{2/3} = 4\), the value is \(\frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12}\).
Power Rule vs. Chain Rule
The power rule applies when the base is plain x. For \((3x+1)^5\) the base is a whole expression, so the answer is not \(5(3x+1)^4\); an extra factor of 3 from the inside is required, which is the job of the chain rule. Likewise, \(2^x\) has a variable in the exponent, so the power rule does not apply to it.
Keep going
Lessons
Worksheets and quizzes
- Power, Sum and Basic Function Derivative Rules – Worksheet
- Power, Sum and Basic Function Derivative Rules – Quiz
Related words
Whole dictionary: A–Z · AP Calculus AB · Calculus