- #1
Miraj Kayastha
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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?
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?
Yes, ln(x) is continuous at negative x-axis. This means that the function has no breaks or jumps in its graph at any point on the negative x-axis.
The domain of ln(x) is all positive real numbers, including 0. This means that the function is not defined for any negative values of x.
No, the derivative of ln(x) is not defined at negative x-axis. This is because the function is not defined for any negative values of x, and the derivative requires the function to be defined at the point where it is being evaluated.
The limit of ln(x) as x approaches 0 from the negative side is undefined. This is because the function is not defined for any negative values of x, so it does not exist at x=0.
No, we cannot extend the domain of ln(x) to include negative values of x. This is because ln(x) is a logarithmic function and the logarithm of a negative number is undefined.