The Power Rule: How to Find a Derivative
Use the power rule d/dx xⁿ = nxⁿ⁻¹ to differentiate polynomials, with two worked examples and the mistakes students make most often.
The power rule
- d/dx xⁿ = n · xⁿ⁻¹
- Bring the exponent down
- Subtract 1 from the exponent
A constant multiplier just comes along for the ride.
Two helpful facts
- d/dx (c) = 0 for a constant
- d/dx (3x²) = 3 · 2x = 6x
- Differentiate each term separately
Sums and differences are done term by term.
Example 1
Differentiate x⁵
- Bring down the 5
- Subtract 1 from the exponent
5x⁴
Example 2
Differentiate 3x² + 4x − 7
- 3x² becomes 6x, and 4x becomes 4
- The constant 7 becomes 0
6x + 4
Common mistake
d/dx x⁵ = x⁴
Do not forget to bring the exponent down as a multiplier.
Check: x⁵ has derivative 5x⁴, not x⁴.
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