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i know that derivative of this function gives us x^n....but is there some theoratical proof also?

like there is a theoratical proof for derivative of x^n

i have a second problem too.....if the integral of a function is 0 then does it necessarily mean that the function is 0?

i read a note written in my class notes that it is not necessarily 0 ...but i can not remember the reason why it is not 0?

i mean if the integral of a function is 0 then it makes perfect sense to say that the function is 0......if this is wrong then please give an example explaining how?

thx