A strange fact as well as big doubt

[iitjee10]

We know x^2 = x*x*x*x... x times
If we differentiate both sides with respect to x, we have
2x=1+1+1+... x times
i.e. 2x=x
i.e 1=2
How is it possible or am i making a mistake?

Please helpp!

 

[Hootenanny]

We know x^2 = x*x*x*x... x times
If we differentiate both sides with respect to x, we have
2x=1+1+1+... x times
i.e. 2x=x
i.e 1=2
How is it possible or am i making a mistake?

Please helpp!

Erm, well to start with x² = x*x

 

[adriank]

I'm assuming you meant x+x+x+x... x times.

The problem with trying to take the derivative of that with respect to x is first, the number of terms changes as well as each term itself, and second, you can't really write it like that if x is not a nonnegative integer.

 

[nicksauce]

And even if x is a nonnegative integer, your function, f(x) = x+x+x... (x times), is only defined at the nonnegative integers, so your function f(x) isn't continuous anywhere, so it can't be differentiated.

 

[lurflurf]

Your way of handling addition and multiplication is confused as you have not prperly defined what x times means. That aside the problem with your differentiation is that for a derivative to be correct all x in the function must be allowed to vary, you hold your x times fixed.
x^2=x+x+x+... xtimes
differentiate
2x=x'+x'+x'+... xtimes
+x+x+x+... x' times
=1+1+1+... x times
+x+x+x+... 1 times
=x+x=2x

So the problems were holding the x in "x times" fixed and not properly defining what "x times means.

 

[nicksauce]

Your way of handling addition and multiplication is confused as you have not prperly defined what x times means. That aside the problem with your differentiation is that for a derivative to be correct all x in the function must be allowed to vary, you hold your x times fixed.
x^2=x+x+x+... xtimes
differentiate
2x=x'+x'+x'+... xtimes
+x+x+x+... x' times
=1+1+1+... x times
+x+x+x+... 1 times
=x+x=2x

So the problems were holding the x in "x times" fixed and not properly defining what "x times means.

Very interesting actually. I had never thought of that.