The discussion revolves around the differentiability of the natural logarithm function, ln(x), particularly at negative values of x. Participants explore the definition of ln(x) and its derivative, considering both real and complex domains.
Participants generally disagree on the differentiability of ln(x) at negative values, with some supporting the traditional real analysis perspective while others introduce complex analysis considerations.
The discussion highlights the limitations of the standard definition of ln(x) in real analysis and the implications of extending the function into the complex domain. There are unresolved assumptions regarding the interpretation of ln(x) in different contexts.
No. If a function isn't defined on some interval, its derivative isn't defined there, either. The real function ln(x) is defined only for x > 0, as you are aware, so the domain for the derivative is x > 0, as well.Miraj Kayastha said:Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?
I assume you are referring to real numbers (you use "x" for the ind. variable and you say that lnx is defined for positive x only) because in the complex field it's all another story...Miraj Kayastha said:Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?