For any real number {{MATH_128}}, {{MATH_129}} {{CMD_130}} We will use logarithmic differentiation. Set {{MATH_131}}. Write {{ENV_132}} Now differentiate both sides, and solve for {{MATH_133}} {{ENV_134}} Whenever {{MATH_135}} is in the domain of {{MATH_136}}, the definition of the derivative at {{MATH_137}} implies that {{MATH_138}}, if {{MATH_139}} and {{MATH_140}}, if {{MATH_141}} , so the power rule applies to {{MATH_142}}, too. Thus we see that the power rule holds for all real-valued (nonzero) exponents. {{CMD_143}} {{CMD_144}}