Question

the following equation is given:
x+x+x+x+x+..+x (x times) = x^2
which is a true sentence.
when we derivative the equation we get:
1+1+1+1+1+..+1(x times) = 2x
which is x=2x lets presume that x does not = 0
and so we get 1=2.
what's wrong with the process that started with a true sentence and ended in a false one???

Answer 1

The issue is with the differentiation on the left-hand side. That function is
\[ f(x) = \sum_{i=1}^x x \]
The first problem is this expression is only defined when x is an integer. But in that case, we don't have continuous function, let alone a differentiable one.

The other problem is even if we did have a differentiable function, the process of differentiation isn't properly respecting the number the x as a summation variable; it is treating that as a constant while differentiating all the terms in the sum, then added up all of those differentiated terms.

Answer 2

(Btw, the obvious consequence of the first problem is also that it is not true for all x that \( f(x) = x^2 \).)

Answer 3

tnk u very much