Euler's constant is one of the most important constant in Mathematics. It comes up everywhere in various fields of Mathematics, such as geometry, finance, and modelling. I find it so incredible, and it has so many properties which I hope to show in this post, and, hopefully, in future posts as well. First of all, let me define it. It is defined to be the following,
The numerical value of this constant is less elegant to show, since it is an irrational number, but here it is, accurate to ten decimal places,
I am going to demonstrate one of the properties of this number that makes it so important in Mathematics.
Firstly, let me introduce exponential functions. They are of the form,
We ignore bases which have a value less than or equal to zero. This is because they are irrelevant to this topic at the moment.
Euler's constant, if used as the base of an exponential form will bring up a remarkable property. It is that it will be it's own derivative. It will satisfy the following,
Indeed, this is true, and I will prove it. But first, let me introduce another function, the natural logarithm. The natural logarithm function is defined to be the inverse function of the exponential function, f(x) = ex . It is usually written as ln(x). To prove that the exponential function, f(x) = ex , is it's own derivative, I must first find the derivative of the natural logarithm function. This will be done using the definition of the derivative,
Applying this to the natural logarithm function, I obtain,
Using the logarithmic identity, I can simplify the limit to the following,
Using a similar identity, I can reduce the limit even further to obtain the following,
Next, I make the substitution
I will then have,
Notice, that now, that the new variable, n, is tending towards infinity. This is because h tends towards 0, and n and h are inversely proportional to each other.
Using the logarithmic identity I used before, the equation will simplify to,
And since, in this case, 1/x is a constant, it can be taken out of the limit like so,
Now, the only thing left is to evaluate the limit. Using the definition of Euler's constant, this becomes very simple. From the definition, we know that,
It should now become very clear what the limit should be. Since n is tending towards infinity, then I can safely conclude that,
Therefore,
Now that I have obtained the derivative of the natural logarithm function, it can easily shown that the exponential function, ex , is it's own derivative.
First of all, let,
Then it follows that,
By the chain rule, differentiating with respect to x will give,
Multiplying both sides by y will give,
And therefore,
This tells us that the derivative of y = ex, is itself, an amazing result.
The numerical value of this constant is less elegant to show, since it is an irrational number, but here it is, accurate to ten decimal places,
I am going to demonstrate one of the properties of this number that makes it so important in Mathematics.
Firstly, let me introduce exponential functions. They are of the form,
We ignore bases which have a value less than or equal to zero. This is because they are irrelevant to this topic at the moment.
Euler's constant, if used as the base of an exponential form will bring up a remarkable property. It is that it will be it's own derivative. It will satisfy the following,
Indeed, this is true, and I will prove it. But first, let me introduce another function, the natural logarithm. The natural logarithm function is defined to be the inverse function of the exponential function, f(x) = ex . It is usually written as ln(x). To prove that the exponential function, f(x) = ex , is it's own derivative, I must first find the derivative of the natural logarithm function. This will be done using the definition of the derivative,
Applying this to the natural logarithm function, I obtain,
Using the logarithmic identity, I can simplify the limit to the following,
Using a similar identity, I can reduce the limit even further to obtain the following,
Next, I make the substitution
I will then have,
Notice, that now, that the new variable, n, is tending towards infinity. This is because h tends towards 0, and n and h are inversely proportional to each other.
Using the logarithmic identity I used before, the equation will simplify to,
And since, in this case, 1/x is a constant, it can be taken out of the limit like so,
Now, the only thing left is to evaluate the limit. Using the definition of Euler's constant, this becomes very simple. From the definition, we know that,
It should now become very clear what the limit should be. Since n is tending towards infinity, then I can safely conclude that,
Therefore,
Now that I have obtained the derivative of the natural logarithm function, it can easily shown that the exponential function, ex , is it's own derivative.
First of all, let,
Then it follows that,
By the chain rule, differentiating with respect to x will give,
Multiplying both sides by y will give,
And therefore,
This tells us that the derivative of y = ex, is itself, an amazing result.
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