As we know that the derivative of $ x^2$ , with respect to $ x$ , is $ 2x$.

i.e., $ \dfrac{d}{dx} x^2 = 2x$

However, suppose we write $ x^2$ as the sum of $ x$ ‘s written up $ x$ times..

i.e.,

$ x^2 = \displaystyle {\underbrace {x+x+x+ \ldots +x}_{x \ times}}$

Now let

$ f(x) = \displaystyle {\underbrace {x+x+x+ \ldots +x}_{x \ times}}$

then,

$ f'(x) = \dfrac{d}{dx} \left( \displaystyle {\underbrace {x+x+x+ \ldots +x}_{x \ times}} \right) $

$ f'(x)=\displaystyle {\underbrace {\dfrac{d}{dx} x + \dfrac{d}{dx} x + \ldots + \dfrac{d}{dx} x}_{x \ times}}$

$ f'(x)=\displaystyle {\underbrace {1 + 1 + \ldots + 1 }_{x \ times}}$

$ f'(x) = x$

This argument appears to show that the derivative of $ x^2$ , with respect to $ x$, is actually x, not 2x..

Where is the error?

 


Error: $x^2$ will equal to $\displaystyle {\underbrace {x+x+x+ \ldots +x}_{x \ times}}$ only when $x$ is a positive integer (i.e., $x \in \mathbb{Z}^+$. But for the differentiation, we define a function as the function of a real variable. Therefore, as $x$ is a real number, there arises a domain $\mathbb{R}- \mathbb{Z}^+$ where the statement $x^2= \displaystyle {\underbrace {x+x+x+ \ldots +x}_{x \ times}}$ fails.

And since, the expansion  $x^2 \neq \displaystyle {\underbrace {x+x+x+ \ldots +x}_{x \ times}}$  for $x \in \mathbb{R}$ , the respective differentiations will not be equal to each other.


Then how can $x^2$ expanded in such a way?

If x is a positive integer:

$x^2= \displaystyle {\underbrace {x+x+x+ \ldots +x}_{x \ times}} $.

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

$x$ 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, $x^2 = [x] \cdot x + {x} \cdot x$

$ x^2 = \displaystyle {\left( {x+x+\ldots +x} \right)_{[x] \, \mathrm{times}}} + x \cdot {x}$

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

$x^2 = x[x]+x\{x\}$

$d/dx {[x²]}= d/dx \left( {x[x] +x \{x\} }\right)$

(differentiation by part)

$= 1\cdot [x]+x \cdot [x]’+ 1\cdot \{x\} + x \cdot \{x\}’$

since $d/dx (x)=x’=1$ and [x]’ & {x}’ represent differentiation of each with respect to x.

$=[x]+\{x\}+x \left({[x]’+\{x\}’ }\right)$

$=x+x (x’)$

$=x+x=2x$

 

  • Yesmanapple sent his view on this article. Have a look.

 

OQhHe

 

Multiplication is not repeated Addition.

  • Greatest Integer Function

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

 


Feel free to ask questions, send feedback and even point out mistakes. Great conversations start with just a single word. How to write better comments?
10 comments
  1. 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.

  2. 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.

      1. $4$ 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.

  3. 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!

  4. 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.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

You May Also Like

Chess Problems

In how many ways can two queens, two rooks, one white bishop, one black bishop, and a knight be placed on a standard $ 8 \times 8$ chessboard so that every position on the board is under attack by at least one piece? Note: The color of a bishop refers to the color of the square on which it sits,…
collatz conjecture
Read More

The Collatz Conjecture : Unsolved but Useless

The Collatz Conjecture is one of the Unsolved problems in mathematics, especially in Number Theory. The Collatz Conjecture is also termed as 3n+1 conjecture, Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, Syracuse Problem. Statement: Start with any positive integer. • Halve it, if it is even. Or • triple it and add 1, if it is odd. If you…
what&#;sthequestionaladin
Read More

What’s the question, if the answer is ‘No!’

Infinitely many answers questions are possible to the answer, “No”. So, our real task should be to find one of THOSE many, which seems to be a perfect one. A simple and the first ever logical approach of giving answers to a question is to derive answers from the question, that is, replace some words of the question with reasonable ones and…

Free Online Algebra and Topology Books

This is a brief list of free e-books on Algebra, Topology and Related Mathematics. I hope it will be very helpful to all students and teachers searching for high quality content. If any link is broken, please email me at gaurav(at)gauravtiwari.org. Abstract Algebra OnLine by Prof. Beachy This site contains many of the definitions and theorems from the area of…
shakunthala
Read More

The Mystery of the Missing Money – One Rupee

Puzzle Two women were selling marbles in the market place — one at three for a Rupee and other at two for a Rupee. One day both of then were obliged to return home when each had thirty marbles unsold. They put together the two lots of marbles and handing them over to a friend asked her to sell then…
featured
Read More

Social Networks for Math Majors

Math or Mathematics is not as difficult as it is thought to be. Mathematical Patterns, Structures, Geometry and its use in everyday life make it beautiful. ‘Math majors’ term generally include Math students, Math professors and researchers or Mathematicians. Internet has always been a tonic for learners and whole internet is supposed to be a social network, in which one…