Find the Error: Differentiation

In summary, the conversation discusses a mathematical riddle involving the derivative of a summation and how it leads to a false result. The mistake is identified as differentiating with respect to a variable that is also a limit in the summation. The conversation also explores the idea of defining the summation for non-integer values and how it affects the differentiation process. The riddle is appreciated for its simplicity and easy-to-remember nature.
  • #1
pbandjay
118
0
This is just a pretty simple "riddle" that I have always liked a lot. I didn't come up with it, I actually got it off of a website a few years ago. I'm sure for some of you, it won't be new, but here goes..

x = x
x = 1 + 1 + ... + 1 (x times)
x(x) = x(1 + 1 + ... + 1)
x2 = x + x + ... + x
D(x2) = D(x + x + ... + x)
D(x2) = D(x) + D(x) + ... + D(x)
2x = 1 + 1 + ... + 1 (x times)
2x = x
2 = 1

Uh oh! What's going on with that real number line? :yuck:
 
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  • #2
I'm going to take a stab at it.

I think the mistake came when you did the derivative. Since your differentiating with respect for x.

2x=(1+1+1 xtimes)+(x+x+x+x)
the ennd result after you take the derivative should be 2x=x+x not 2x=x
 
  • #3
The problem arises when you take the derivative of

x+x+...+x

and get

1+1+...+1

You see, the 1+1+...+1 can be represented as a summation over j with limits 1 and x. But since you differentiate with respect to x, and x is one of the limits, you can't just differentiate the summand.
 
  • #4
That is a cute one. :tongue2:

But... (1 + 1 + ... + 1) "x times"?
The right hand side is only defined for non-negative integer values of x.
How can you write the symbol 1 "x times" when x is 1.5, or square root of 2, or pi? How can you add 1 to itself "x times" when x is pi? If you cannot define this for all real numbers, then the function is not from R to R and is therefore not differentiable (in the sense of real functions.)
We see the problem more clearly if we differentiate earlier in the process:
x = 1 + 1 + ... + 1
D(x)=D(1 + 1 + ... + 1)
1 = D(1) + D(1) + ... + D(1)
1 = 0

You must be able to effectively compute the action 1 + 1 + ... + 1 "x times".
You could try to defined it for non integer values.
But if you define it as meaning x, that is
1 + 1 + ... + 1 "x times" = x,
then that is the effective form that we would differentiate. To take the derivative of this new version we would first need to write it in its effective form. So, in this case we get D(1 + 1 + ... + 1 "x times") = D(x) = 1 by definition.
 
  • #5
For fun abuse of notation,

D(x + x + ... + x (x times)) = (1 + 1 + ... + 1) (x times) + (x + x + ... + x) D(x times)
 
  • #6
Hurkyl said:
For fun abuse of notation,

D(x + x + ... + x (x times)) = (1 + 1 + ... + 1) (x times) + (x + x + ... + x) D(x times)

:approve: Which, of course, we have D(x times) = (1 times), so...

D(x + x + ... + x (x times)) = (1 + 1 + ... + 1) (x times) + (x + x + ... + x) D(x times)=

(1 + 1 + ... + 1) (x times) + (x + x + ... + x)(1 times)= x + (x) = 2x
 
  • #7
Haha I should have known you all would have chopped this up so quickly! The reason I like this one so much is because it is very easy to remember, too.
 
  • #8
so I'm just wondering. was what i posted right? lol :p

I haven't done this stuff in like 2 years nearly.
 

1. What is differentiation?

Differentiation is a mathematical process used to find the rate of change of a function at a specific point. It involves finding the derivative of a function, which represents the slope of the tangent line at that point.

2. Why is differentiation important?

Differentiation is important because it allows us to analyze the behavior of functions and understand how they change over time. It has many real-world applications, including in physics, economics, and engineering.

3. What are the basic rules of differentiation?

The basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule. These rules allow us to find the derivative of more complex functions by breaking them down into simpler parts.

4. How do you find the error in differentiation?

In most cases, the error in differentiation is caused by mistakes in applying the basic rules or algebra. It is important to carefully check each step of the process and use the correct rules and formulas. Sometimes, using different methods of differentiation, such as the limit definition, can also help identify errors.

5. How can I improve my skills in differentiation?

The best way to improve your skills in differentiation is by practicing regularly. Start with simple functions and gradually move on to more complex ones. It can also be helpful to seek guidance from a teacher or tutor, and to use online resources and textbooks for extra practice and review.

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