power rule

Most of the functions you meet are built from simple powers of x — x squared, x cubed, x to the one-half (a square root), even x to a negative power (a reciprocal). The power rule is the quick shortcut for differentiating every one of them, so you never have to crank through the limit definition by hand. It's the first rule everyone learns and by far the most used.

The rule says d/dx[x^n] = n x^(n-1). In words: bring the exponent down in front as a multiplier, then knock the exponent down by one. So the derivative of x^3 is 3x^2; of x^5 is 5x^4; of x (which is x^1) is 1; and of a constant (x^0) is 0. It works for any real exponent n, not just whole numbers: the derivative of sqrt(x) = x^(1/2) is (1/2)x^(-1/2), and the derivative of 1/x = x^(-1) is -x^(-2) = -1/x^2.

Two cautions keep people out of trouble. First, the power rule is for a variable raised to a constant power, like x^n — it does not apply to a constant raised to a variable power, like 2^x, which needs a different rule. Second, when a coefficient is attached, it just rides along: the derivative of 5x^3 is 5 times the derivative of x^3, namely 15x^2, because constant multiples factor straight out.