A proof of the power rule by Taylor's theorem

Abstract

The standard way of proving the power rule for irrational exponents is to use the differentiation of exponential and logarithmic functions. We derive the power rule for irrational exponents from that for rational exponents by using Taylor's theorem, without relying on exponential and logarithmic functions. This approach is straightforward in view of extending the definition of the power function from rational exponents to irrational exponents. It also helps students to understand that Taylor's theorem is useful to investigate the local behaviour of functions.