> For the complete documentation index, see [llms.txt](https://julienbeaulieu.gitbook.io/wiki/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://julienbeaulieu.gitbook.io/wiki/sciences/math/calculus/the-big-picture-of-calculus/the-exponential-e-x.md).

# The exponential e^x

### Special properties

Its slope is equal to the function. (reminder - slope : rate of change over a very small distance)

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfaAlggK19rbHBGlM9a%2F-LfaIwBATT_tzzb1oaEU%2Fimage.png?alt=media\&token=447b0455-a2f3-4391-9d33-4a054bd288ed)

Climbs much faster than y=x^100 for instance. The highest y gets, the higher the slope, so it keeps increasing. 

Where does this happen? Interest. The more money you have the more interest, so more money, more interest, etc. 

Let's construct it: 

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfaAlggK19rbHBGlM9a%2F-LfaKht0iFOKO3al3LD6%2Fimage.png?alt=media\&token=4f3f778c-b1ff-401d-b908-00bbcdf51178)

y(x) = dy/dx by definition of e^x. Also what we're adding becomes smaller and smaller because n! grows faster than x^n. So it doesn't keep growing, it stops at some point. 

Now let's check why e^xe^X = e^(x+X)

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfaAlggK19rbHBGlM9a%2F-LfaLSXPDH6PikZxoEI3%2Fimage.png?alt=media\&token=fd2c07a3-4030-4a6d-bcba-765b9f07a9d4)

It works. 

So with this we can find out what e (euler's number) is:for x=1 we have

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbVzPb_qqc_4VbdJSp%2F-LfbWrfzUVp2sRqUm7Bo%2Fimage.png?alt=media\&token=00438b40-966b-418d-ac94-2a3ce765acb3)

It's not really 2.7, 2,71828......

If we graph it: 

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbVzPb_qqc_4VbdJSp%2F-LfbXJ9KOvkMGi93ONV_%2Fimage.png?alt=media\&token=f8a52c42-766e-41e1-a5ad-dbd7919ee779)

Let's have an example: computing compound interest. 

Say we have 1 dollar, and the bank gives you 100% interest after 1 year. we you get, 2, 4, 8 ... $. 

If you ask the bank to give you interest after each month, you'd get 1+(1/12)(1/12)^2..., every day = 1/365. But we want continuous interest because we're doing calculus, so N.

When N tends towards infinity, (1+1/N)^N is e. 

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbVzPb_qqc_4VbdJSp%2F-LfbYKIK5y4G0W_Owt4T%2Fimage.png?alt=media\&token=18fc5066-f07d-49e4-8724-d1cbce518e48)

If we add c constant - 

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbVzPb_qqc_4VbdJSp%2F-LfbZ5s-hQG4YxcfiU4i%2Fimage.png?alt=media\&token=aea88418-a8cf-4922-a864-5fb18775ab00)

(property: The slope of the tangent line to any point of the graph below equals the height of that point above the horizontal axes. This means that y=e^x = y'=e^x too. 

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbPL4qkB8NvjQ0uec7%2F-LfbPPPyb-FI-8cCsbUg%2Fimage.png?alt=media\&token=d0f0dcd3-ce35-407d-a3af-07adaa843e5a)

These are equivalent:

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LfbPL4qkB8NvjQ0uec7%2F-LfbQbmN6QJWou6cXvMX%2Fimage.png?alt=media\&token=3a1f2321-5d6d-4feb-8f32-9422c19eecd5)

2^t = e^ln(2)\*t. That is why we almost never see anywhere a^t, we can always write it e^ct.