2.5 The Power Rule

The rule

d/dx[x^n] = n·x^(n−1) for any real exponent n: bring the exponent down as a factor and subtract 1 from it. Examples: d/dx[x³] = 3x², d/dx[x] = 1.

Roots and reciprocals

Rewrite roots and reciprocals as powers first: √x = x^(1/2) so d/dx[√x] = (1/2)x^(−1/2); 1/x = x^(−1) so d/dx[1/x] = −x^(−2).

Key terms

Power rule

d/dx[x^n] = n·x^(n−1).

Example

d/dx[x³] = 3x².

d/dx[x]

Equals 1.

Roots

Rewrite √x as x^(1/2) before differentiating.

Reciprocals

Rewrite 1/x as x^(−1); d/dx[1/x] = −1/x².

Try an AP-style question

d/dx (1/x³) = ?

1. 3x⁻⁴
2. −3x⁻⁴
3. 1/(3x²)
4. −3x⁻²