At a point, the derivative is defined to be. Note for second-order derivatives, the notation is often used. These are called higher-order derivatives. Therefore, it is proved that the derivative of logarithm of a function with respect to variable is equal to the product of the reciprocal of the function and the derivative of the function. When a derivative is taken times, the notation or is used. It’s easiest to see how this works in an example. Anyway, I think the example of -x2 isn’t a good example for a log transformation, since we take logs of positive numbers, but take the log of the function for perhaps the normal distribution to get a sense of how the maximizing input is the same with and without the log.
This is called logarithmic differentiation. Now that we know how to find the derivative of log(x), and we know the formula for finding the derivative of log a (x) in general, lets take a look at where this formula comes from. The derivative of logarithmic function can be derived in differential calculus from first principle. Taking the derivatives of some complicated functions can be simplified by using logarithms.