What you're actually doing there is differentiating kx to get k.

]]>But the number of terms changes with x, so the equation keeps changing, plus for negative x or fractional x's,

how many terms are there?]]>

x+x+x+.....+x (x times) is only defined for x in

So that is isn't continuous anywhere or differentiable, really...

]]>The derivative of x², with respect to x, is 2x. However, suppose we write x² as the sum of x x's, and then take the derivative:

Let f(x) = x + x + ... + x (x times)

Then f'(x) = d/dx[x + x + ... + x] (x times)

= d/dx[x] + d/dx[x] + ... + d/dx[x] (x times)

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

= x

This argument appears to show that the derivative of x², with respect to x, is actually x. Where is the fallacy?

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