The derivative of ln(x) is 1/x.
This is because ln(x) is the inverse of the exponential function, e^x. Therefore, the derivative of ln(x) is the inverse of the derivative of e^x, which is 1/x.
To prove the derivative of ln(x) using the definition of a derivative, we start by writing out the definition: f'(x) = lim(h→0) [(f(x+h) - f(x)) / h]. Then, we substitute ln(x) for f(x) and simplify the expression until it becomes 1/x.
Yes, the derivative of ln(x) can be negative for values of x less than 1. This is because the natural logarithm function is only defined for positive values of x and its derivative is negative for values less than 1.
The derivative of ln(x) represents the slope of the tangent line to the graph of ln(x) at any given point. Therefore, the graph of ln(x) and its derivative have a reciprocal relationship, where the derivative is the inverse of the slope of the graph of ln(x).