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x2

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

Sunday, February 6th, 2011 12:01 / 5 Comments

mathematical beauty via flickr.com

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?

(more…)

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