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当使用 Δx 进行区分时,会发生以下的转换


f g 可得


对于乘f g 的导数,我们得到的是


而这无论如何都会变成 0。因此,积规则

f g )' = f′ g + f g′



Small increments

When calculating the number e we use the formula

and see how you can get the correct result. We start with usual numbers

and see that the outcome increases during each step, although the value of the fraction decreases steadily. Eventually you work with infinitely small terms, but you must not neglect those here.



The differential of the logarithm gives

Subtraction of logarithms produces

Substitution of this in the power series for the logarithm gives

Because all the differentials of the second order and higher are neglectable, you may write

After substitution you get


The number 1

You can write the number 1 with an infinite number of decimals as

The three points indicate that there are infinitely many decimal places. You can calculate that with a fraction

It is said that 0.999999… approaches 1 in infinity. That sounds impressive, but no one can imagine what infinity is. We feel that there must be a neglectable difference between 0.999999… and 1. That's however wrong. It are just two different ways in which the same number can be written.



In principle an infinitely small value in a calculation may be neglected. If it occurs infinitely often, however, it should not. That is a rule of thumb. We know that

And then you may certainly not neglect

∞ × ∆x = ∞

In all calculations you should strictly apply the mathematical rules. Therefore you must use limits in these cases, because then you know what you are doing. Where necessary to avoid confusion you write


In itself it is obvious when you can neglect an infinitesimal value.


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