The natural logarithm function, ln(x), is strictly defined for positive values of x, making it differentiable only in that domain. The derivative of ln(x) is 1/x, applicable solely for x > 0. While the function 1/x is defined for all x ≠ 0, it does not represent the derivative of ln(x) outside its defined domain. In complex analysis, ln(x) can be extended to negative values using the expression ln(-x) + πi, but this does not affect its differentiability in the real number domain.
PREREQUISITESMathematicians, students studying calculus, and anyone interested in the properties of logarithmic functions and their differentiability across different domains.
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?