Key takeaway: The power rule states that the derivative of x raised to the number n is n times x raised to n minus 1, and it works for whole, negative and fractional exponents.
Grade 12 – AP Calculus ABAP Calculus AB Unit 2Calculus
Definition of Power Rule
The power rule is the quickest way to differentiate a variable raised to a constant exponent. Bring the exponent down in front as a multiplier, then lower the exponent by one.
It is valid for every real exponent, so roots and reciprocals are covered once they are rewritten with exponents: a square root is the power \(\frac{1}{2}\), and \(\frac{1}{x^2}\) is the power \(-2\). A lone \(x\) has exponent 1, so its derivative is 1.
Formula and notation
For any real number \(n\):
\[ \frac{d}{dx}(x^n)=n\,x^{n-1}. \]
With a constant multiplier \(a\), the rule becomes \(\frac{d}{dx}(a x^n)=a n\,x^{n-1}\). The base must be the variable itself, and the exponent must be a constant.
Key facts
- A constant term has exponent 0 attached to x, so its derivative is 0.
- Negative exponents work the same way: x to the -3 differentiates to -3 times x to the -4.
- Write radicals and fractions as powers first, then differentiate each term.
- The rule does not apply to constant^x, such as 2^x, where the variable sits in the exponent.
Power Rule example
Find the slope of \(y=5x^3+\frac{8}{x^2}\) at \(x=2\).
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Rewrite with exponents: \(y=5x^3+8x^{-2}\).
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Apply the power rule to each term: \(y^{\prime}=15x^2-16x^{-3}\).
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Substitute \(x=2\): \(15\cdot 4-\frac{16}{8}=60-2=58\).
The slope of the curve at \(x=2\) is 58.
Common mistake
Watch out
A student differentiates \(x^{-2}\) and writes \(-2x^{-1}\), lowering the exponent by moving it toward zero.
Why it is wrong
The rule subtracts 1 from the exponent, and subtracting 1 from \(-2\) gives \(-3\), not \(-1\). The correct derivative is \(-2x^{-3}\). Write the subtraction out, \(-2-1=-3\), whenever the exponent is negative.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus