Differential coefficient of $x^2$ if $x$ is a positive integer

We all know that the differential coefficient of $x^2$ is $2x$ . But if we write
$x^2 = \underbrace{x+ x+x+x+\cdots + x}_{x \text{ times}}$
On differentiating both sides with respect to $x$ we get: $d(x^2)/dx = \underbrace{1+ 1+1+1+\cdots + 1}_{x \text{ times}}$ = $x$
Why has the differential coefficient changed?

Similarly one can find differences in differential coefficients of $x^3$ , $x^4$ , $x$ ,etc. if $x$ is a positive integer.

Please help in finding my mistake.

#Calculus

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Comments

First off, note that equation is valid only if $x$ is an integer. As the equation doesn't hold in any neighbourhood of a positive integer, you cannot differentiate the given equation on both sides!

Hence, the result of differentiating that equation is wrong!

A Former Brilliant Member - 4 years, 11 months ago

That's right. It is just like differentiating xy wrt x and then substituting y=x.

Aditya Kumar - 4 years, 11 months ago

Nice example.

A Former Brilliant Member - 4 years, 11 months ago

It's there in there

Viki Zeta - 4 years, 11 months ago

Thank you I found my mistake.

Priyamvad Tripathi - 4 years, 11 months ago

If you look at it correctly, one way of interpreting is that the sum is only valid for for integer values. But another way of seeing it is that in writing the sum the numbers of times is x occurs is also a function in x, 'x times'. So it is possibly an implicit function of two variables. Like sum x y number of times. Now you can differentiate it wrt to x.

Abhi Kumbale - 4 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Differential coefficient of x2x^2x2 if xxx is a positive integer

Comments

Differential coefficient of $x^2$ if $x$ is a positive integer