A mathematical multiplying puzzle

Hi all,
Here is a puzzle I hope most of you know.... For those who don't, but want to tease their own head.......
We know 5*6 = 5+5+5+5+5+5 i.e. 5 is added 6 times.
Similarly 5*4/3 = (5+5+5+5)/3 i.e. 5 is added 4 times and thendivided by 3 which is equivalent to summing 5 4/3 times or summing 5/3 4 times. In general for all rational numbers we can write
x * y = x+x+x+x+.....upto y times or y+y+y+y+....upto x times. Following this logic
x² = x * x = x+x+x+x+....... upto x times
Tose who are suspecting, I tell you there is no trick in this. It is mathematically correct. The puzzle starts now. Let's differentiate both sides. We get from left side 2x and from right side we get 1+1+1+1+....upto x times. This meane
2x = 1+1+1+1+....upto x times
i.e. 2x = x
or 2 = 1 as x is not bound to be zero. Definitely something is wrong in it. Where is it?

Re: A puzzle

you can't differentiate and assume x to be a constant (x times x) at the same time.

Re: A puzzle

Interesting!

(d/dx)x²=(d/dx)x+...

is it true this condition applied for diff eq?

Does this apply for integral too?

∫x²dx=∫xdx+...

actually, it cute!

Re: A puzzle

no interesting~：）
x*x=x+x+......+x
in this, x is integer
but, we should know, the define of derivative~
the dx should be infinitesimal~

A puzzle

C'mon guys...just play stupid so this topic can go way uncontrolled.

It's cute...really cute looking case.

Is it (d/dx)x!=(d/dx)x*(d/dx)(x-1)*...*(d/dx)(1)?