Power Functions: Properties and Derivatives

This lecture covers the properties of logarithms in base b, including the logarithm of 1, the relationship between different bases, and the power rule for logarithms. It also introduces generalized power functions defined as the exponential of the product of the natural logarithm of a function and another function. The lecture further explores the derivatives of logarithmic and exponential functions, focusing on the Newton quotient and the derivative of the natural logarithm. Examples are provided to illustrate the concepts, such as the derivative of exponential functions and the behavior of sine times cosine functions. The lecture concludes with the application of theorems to determine the sign of derivatives and the calculation of exponential derivatives.