Why is the derivative of the exponent 0?

Greetings, dear Readers! I am sure that many of you from your school mathematics course remember perfectly a wonderful function – an exponent, the derivative of which, no matter how much you take it, is equal to the original function.

  The number at the base of the exponent function is the famous Euler's number e = 2.718281828 ...
The number at the base of the exponent function is the famous Euler’s number e = 2.718281828 …

However, how many of you know why this is happening? Today I want to tell this in the simplest possible language. Go! Consider two exponential functions:

Let us now recall the classical definition of the derivative of a function as the limit of the ratio of the increment of a function to the increment of its argument when the increment of the argument tends to zero.

  • In simple words: we analyze the rate of change of the function f (x) with an infinitesimal change in its argument, which we will denote by ∆x.

In the formulas for the first function, it looks like this:

Let’s calculate something on the calculator, namely the expression under the limit sign. For example, let the change in the function ∆x = 0.001. Then:

However, this will not give us anything … Until we calculate a similar expression for a function based on 3:

But this is already interesting. If we recall a little mathematical analysis, then it pops up in my head second Bolzano-Cauchy theorem or intermediate value theorem

In our case, it allows us to assert that the function under consideration (meaning the fraction (x ^ ∆х-1) / ∆х) for some x equals one! If we find such an x, then by definition we obtain the equality of the function of its derivative! Begin! We equate our function to one:

It's ... just ... an exclamation point
It’s … just … an exclamation point

The second remarkable limit is the ratio known from the school course, which invariably leads to the Euler number. So the proof is over!

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