Oldie but goodie 1=2

[bonanova]

You can multiply by adding, so you can make squares that way, too.

• 1² = 1
• 2² = 2+2
• 3² = 3+3+3
• 4² = 4+4+4+4
• ...
• x² =  x+x+x+x+ ... +x (x times)

The derivative is

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

2 = 1

What's wrong here?

 

[rocdocmac]

Spoiler

You cannot use the sum rule of differentiation to determine the derivative of  x + x + x + x + x … (repeated x times) since the x in “x times” is not a number, but a variable. The sum rule in differentiation is

(f₁(x) +  f2(x) + ... + fk(x))'  = f  '₁(x) +  f  '₂(x) + ... +  f 'k(x), where k is any positive integer.

 

[bonanova]

Doing the derivative correctly,

Spoiler

Define f(x) = sum _(i=1,x) x.

Then f(x+1) = sum _(i=1,x+1) (x+1)

And f '(x) =~ { f(x+1) - f(x) } / 1 = { (x+1)² - x² } / 1 = ( 2x + 1 ) / 1  ~  2x (when 1 is small compared to 2x)

So the derivative of the last sum is 2x, not x.