Extremely puzzling math question , that I can't figure out, plz help me, o wise ones!!!?

Question

Spoiler alert: May involve basic calculus, but nothing you can't handle.



You can write X-square as X added to itself X times, Right?

As in 5^2 =5+ 5 + 5+ 5 +5=25. Okay?

I mean

Let Y = X^2 = X + X + X +X +... X times,

But if you differentiate the two expressions with respect to X

if you take Y=Xsquare, differential is dY/dX = 2X

and otherwise,
its
dY/dX=1+1+1+1+.....1 X times = X


But this curve can have only 1 tangent at a point, it is a parabola , right?

So if the slope is same,

2X = X
=>2=1!!!!!

Howzzat????

Well, i need to clarify here!

Hi when I say Xsquare = X + X + X+ ... X times,

I mean

1square= 1+ ... 1 times = 1
2 square= 2+... 2 times =4

3^2 = 3 + 3 + 3 =9

4^2 = 4 + 4 + 4 +4 =16

5^2 = 5 + 5 +5 +5+ 5 = 25

6^2 = 6 + 6 + 6 + 6 + 6+ 6= 36

7^2 = 7+ 7 + 7 + 7 + 7 + 7 + 7 = 49

8^2 = 8 + 8+ 8 + 8 + 8 + 8 + 8 + 8= 64

And so on. Do you get the point now?

Thus

X^2 = X + X + X + X + X+ X + X+ ... X times

Answer 1

Of course, your two functions aren't the same at, say, X = 6

6^2 = 36 and 6+6+6+6+6 = 30.

Nice try though.

The only places where your two functions take on the same value is at X = 5 and X = 0. Since they are not equal everywhere, of course they would not have the same derivative.

edit: jfengel, my reasoning is fine. I'm stating that saying that x+x+x+x+x is somehow the same as x+x+x+x+x+x is not correct; one is 5x and the other is 6x, and both are certainly not equal to x^2. They are different functions, therefore they have different derivatives. Just because my reasoning is different from yours doesn't make what I said any less true.

Answer 2

Very clever. Previous answers are incorrect; you've expanded the exponentiation and multiplication correctly.

The problem is that the rule for differentiating a function doesn't allow you to do that. It only applies to a fixed set of additions, not to the variable set of additions you've thrown at it. That's even more apparent when you consider a function like sqrt(x) = x^.5; you can't product a series term like that with a non-integer exponent.

There is a rule that d(f(x)+g(x))/dx = d(f(x))/dx + d(g(x))/dx, but that rule doesn't apply in this case because you've got an arbitrary number of additions. You're tempted to, because ordinarily addition is commutative (i.e. a+(b+c) = (a+b)+c)) so you just group the first two and iterate until you've done all of them. But since you don't know how many there are, you can't do that.

If you tried in this case, you'd get x^2 = X + (X+X...) X-1 times, which equals X + X*(X-1). If you differentiate that, you get 1 + 2X - 1 = 2X. Essentially, you've derived it by induction.

Answer 3

X^2 is not X+X its' X*X

Answer 4

Great problem! The difficulty is obviously in the "You can write X-square as X added to itself X times, Right?"

WRONG!!!

The problem here is that X is continuous, not discrete!
You are only accounting for a few discrete points on the axis, those being integers! What about X= 1.335? What about X=sqrt(2)? The way you have expanded X^2 should actually differentiate according to the product rule for X taken X times.

X taken X times = X*X
d(X*X)/dx = Xdx/dx + X dx/dx = 2X

Answer 5

jfengel's answer is correct. To look at it another way, your expression for x^2 can be written

x
∑ x
1

Now you want the derivative of that:
........x
d/dx ∑ x
........1
What you are doing is ignoring the fact that the variable x appears in two parts of the function: the argument of the sum and the upper limit of the sum. You are only differentiating the argument, ignoring the limit as a variable.

Answer 6

well obviously 1+1+1+1 x times is going to be (1+1)x=2x
So 2x=2x
x=x
2=2
ta-da!

and to whoever answered first, you have to add the number x times, so if x=6 you add 6 to itself 6 times, not 5, and 6+6+6+6+6+6=36

Answer 7

hey, x^2 is not x +.... x .
it's x multiplied by x.
and you can say multiplication is adding like 2 * 1 =2.
but, how can you do x times x ,?
so, if you can find, x times x , either by using anything like AP or GP. than, we can continue differntiating that! huh?

Answer 8

I think that jfengel just proved what he said wasn't true:

1 + 2x - 1 = 2x is a completely true equation.