textbook-theory · English
General Power Functions
A lecture note covering the definition and differentiation of general power functions of the form f(x) = x^a for any real number a. The document establishes the identity x^a = e^(a ln x) for positive x, then systematically defines the function for x = 0 and negative x according to the rational or irrational nature of the exponent. The derivative formula d/dx(x^a) = ax^(a-1) is derived with analysis of where the derivative exists, illustrated by several Chain Rule examples.
power functionsdifferentiationcalculuschain rule