Generalized Powers: Derivatives and Exponentials

This lecture covers the properties of logarithms in base b, generalized powers, and the derivatives of logarithmic and exponential functions. It also explores the concept of exponentials as the inverse of logarithms, providing examples and discussing the behavior of functions under differentiation. The lecture delves into the calculation of derivatives using the Newton quotient and the application of the squeeze theorem to determine the derivative of the natural logarithm. Additionally, it examines the derivative of the exponential function and the relationship between sine and cosine functions through exponentiation. The lecture concludes with a detailed analysis of the derivative of the sine-cosine function and its behavior under differentiation.