Differential coefficient of x2x^2 if xx is a positive integer

We all know that the differential coefficient of x2x^2 is 2x2x. But if we write
x2=x+x+x+x++xx timesx^2 = \underbrace{x+ x+x+x+\cdots + x}_{x \text{ times}}
On differentiating both sides with respect to xx we get: d(x2)/dx=1+1+1+1++1x timesd(x^2)/dx = \underbrace{1+ 1+1+1+\cdots + 1}_{x \text{ times}} = xx
Why has the differential coefficient changed?

Similarly one can find differences in differential coefficients of x3x^3,x4x^4,xx,etc. if xx is a positive integer.

Please help in finding my mistake.

Note by Priyamvad Tripathi
5 years ago

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1 vote

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First off, note that equation is valid only if xx 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!

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

Aditya Kumar - 5 years ago

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Nice example.

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Thank you I found my mistake.

Priyamvad Tripathi - 5 years ago

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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, 11 months ago

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