I was playing around, and I was wondering, is this a true statement?
$$\frac{ d^n }{ dx^n }(x^n)=\int\limits_{0}^{\infty}x^n e^{-x}dx$$

Question

I was playing around, and I was wondering, is this a true statement?
$$\frac{ d^n }{ dx^n }(x^n)=\int\limits_{0}^{\infty}x^n e^{-x}dx$$

kainui · Answer

Anyone know a better place to ask math questions like this? No one ever responds to my questions here lol.

xishem · Answer

I'd go to a forum to ask some of the higher-level questions. I like http://www.physicsforums.com/, because it has physics and math sections as well as many other subject sections.

kainui · Answer

@Xishem Thanks, I appreciate it!

kc_kennylau · Answer

Well I think that $\dfrac{d^n}{dx^n}x^n=n!$

kainui · Answer

Yeah, it totally does, but is it true for fractions also? @kc_kennylau

kainui · Answer

http://www.wolframalpha.com/input/?i=%281%2F2%29%21%3Dgamma+function+%281%2F2%29

kc_kennylau · Answer

Well can you differentiate an expression half times?

kainui · Answer

Yes, actually. http://www.amazon.com/Introduction-Fractional-Calculus-Differential-Equations/dp/0471588849

kainui · Answer

For instance,

$$\frac{ d^n }{ dx^n }(e^{ax})=a^n e^{ax}$$

kc_kennylau · Answer

Wow can you teach me lol (just give me one example of differentiating a function half time and i may get it)

kainui · Answer

Oh see I don't understand it, I'm just playing with it because I had an idea, what if you could do it? Then I found out it was a real thing. You can also take imaginary, pi, or whatever derivative you want, not just fractions. Weird, but I really don't understand it. Another fun thing I discovered was if you take the derivative of a sine or cosine function:

$$\frac{ d }{ dx }(\sin(x))=\cos(x)=\sin(x+\frac{ \pi }{ 2 })$$
then you can see how this really just becomes:
$$\frac{ d^n }{ dx^n }(\sin(x))=\sin(x+n\frac{ \pi }{ 2 })$$

kinda weird and interesting, but if you do linear transformations to get there with matrices, you'll also find that you can find the square root of the derivative matrix to get the same answer, kinda cool haha.

kainui · Answer

Also, if you do a negative derivative of sine like this, it gives you consistent answers for integrals of sine also. Cool huh?

loser66 · Answer

as you know, the right hand side = n!, 
Now, calculate the left hand side: it's improper integral, so, we have to convert it to lim. 
Let say |dw:1385728662267:dw|