Originally Posted by

**DrRocket**
Which is WRONG unless f is non-negative.

It is not "wrong," it just lead to a complex solution for all f(c) < 0. This is not a real-valued function. You might as well say that the power rule is wrong for n = 1/2 for x negative.

Originally Posted by

**DrRocket**
This makes little sense.

It means that the rules of differentiation should be as such:

product rule: f(x)g(x)

chain rule: f(g(x))

quotient rule: f(x)/g(x)

power rule: f(x)^g(x)

because those rules are more general as opposed to those used today (which are the same except for the power rule being for x^n).

Originally Posted by

**Ellatha**
ridiculous

Why? For any function f(x) one can use the difference quotient, yes, but by that method one would have to find the derivative rather than using the formula provided (which is a matter of plugging in).

Originally Posted by

**DrRocket**
There is no good vreason to do this.

I've already addressed this.

Originally Posted by

**DrRocket**
You are indeed chasing your tail and making something that is quite simple look complicated. That is the opposite of good mathematics.

The work required to get to the formula is sophisticated, but the results (which are what are important), are rather simple, all they state is the following:

If anything, the current power rule may still be called the power rule, but the f(x)^g(x) method that I mention should be considered a generalized (and overall better) version of it. This is because it makes functions that would be extremely tedious to differentiate much less so (as they would require the chain rules as well as a variety of others). On the other hand, I will concede that slightly more simple functions, such as the case of x^n, require a little more work, but by that token one can simply derive a formula for such simple cases (as I already have shown by proving the current power rule).