Power Rule for Derivatives: Rational Exponents - Expii
If r=p/q is a rational constant, then the derivative of x^r is rx^(r-1) whenever x^r is defined. (When p=1, this tells you how to differentiate the q-th root function x^(1/q).) One way to derive this is to write x^r = g(f(x)) where g(y) = y^p and f(x) = x^(1/q) (a radical), and use the chain rule, power rule for integer exponents, and derivative of the q-th root. Another way is to write x^r=e^(r ln(x)) and use the chain rule, exponential rule, and logarithm rule.