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# Derivative of x squared is 2x or x ? Where is the fallacy?

Home / Derivative of x squared is 2x or x ? Where is the fallacy?

As we know that the derivative of , with respect to , is

.

i.e.,

However, suppose we write as the sum of ‘s written up times..

i.e.,

Now let

then,

This argument appears to show that the derivative of

, with respect to

, is actually x, not 2x..

Where is the error?

Error:

will equal to

only when

is a positive integer (i.e.,

. But for the differentiation, we define a function as the function of a real variable. Therefore, as

is a real number, there arises a domain

where the statement

fails.

And since, the expansion

for

, the respective differentiations will not be equal to each other.

## Then how can // < ![CDATA[ // < ![CDATA[ x^2 // ]]&gt; // ]]> expanded in such a way?

If x is a positive integer:

.

But when when x is an arbitrary real number >0, then

can be written as the sum of it’s greatest integer function [x] and fractional part function {x}.  (See this video for more details.)

Therefore,

So, we can now correct the fallacy by changing the solution steps to:

(differentiation by part)

since

and [x]’ & {x}’ represent differentiation of each with respect to x.

• Greatest Integer Function

(last updated on 13th December 2013, 12:45:17 PM IST)

• Luitzen Hietkamp / November 4, 2011 / Reply

You simply failed to take account of the fact that not only the value of x changes, but also the size of the set itself, which you didn’t. In reaction to the second reply:

x² = xW(x)+xF(x) Why not just write x² = xW(x) = x*x ? Then you can differentiate this by parts as well.

And why isn’t multiplication repeated addition? The blog only says it isn’t, without explaining why. As far as I know, multiplication is repeated addition. This fact is very useful if you need to multiply long numbers, like 1,345,843 *3,464,901, in your head or with paper.

• Gaurav Tiwari / November 4, 2011 / Reply

Hi! Thanks for your comment. $x^2 =x+x+x+\ldots +x$ is true, if and only if x is a positive integer.
But x*x is as same as:
x*x =x*([x]+{x})
where [x] is integer part of x and {x} is fractional part of x. This post is very old and it need to be edited since I had used W(x) and F(x) for [x] and {x} respectively.

Regarding, multiplication is not repeated addition: How can you explain— ${5.74}^2$, or ${-4}^2$ as addition? One can’t add any number fractional number or negative number of times.

• j / April 2, 2012 / Reply

4^2 = 4 * 4 = 4 + 4 + 4 + 4

• Gaurav Tiwari / April 2, 2012 / Reply

$4$ is a fixed positive integer. You can add things upto 4 times, but not all $x \in \mathbb{R}$. Differentiation, here, is defined on real numbers.

• Raja / December 13, 2013 / Reply

Its obvious..The fault is in the beginning itself..Why you are making very absurd assumption.
You cannot write $x^2=x+x+x+\ldots$, but you can write $x^2=x+x$.
How can you say
“However, suppose we write $x^2$ as the sum of x ‘s written up x times..” If it your assumption, then it is not $x^2$..actually it is for $x^x$.
Got it!

• (Author) Gaurav Tiwari / December 13, 2013 / Reply

Hmmm. Weird comment. $x+x$ is $2x$ not $x^2$.
And, $x^x$ means $x$ multiplied to itself $x$ number of times.

• Raja / December 13, 2013 / Reply

I think better change to:
Derivative of x squared is 2 ? Where is the fallacy?
Yes x^x means x multiplied to itself x number of times
and x^2 means x multiplied to itself, i.e x times x or X x X.
But you say x^2 means x multiplied to itself x number of times.

• (Author) Gaurav Tiwari / December 13, 2013 / Reply

No. I have written, $x^2$ as the sum of x‘s written up x-times, not that

x^2 means x multiplied to itself x number of times

.
Please read the article once again and http://www.maa.org/external_archive/devlin/devlin_01_11.html <– this one too.

• Ankur / July 15, 2014 / Reply

do u even know the ‘M’ of mathematics? :p