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# Tag Archives: Variable

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

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?