using limits show that derivative of x^2 is 2x.

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Let us say;

`y = x^2 ---(1)`

Then for a small change `deltax ` in x if we have `deltay` change in y;

`y+deltay = (x+deltax)^2 ------(2)`

(2)-(1)

`deltay = (x+deltax)^2-x^2`

`deltay = (x+deltax+x)(x+deltax-x)`

`deltay = (2x+deltax)(deltax)`

The derivative of y is defined as;

`(dy)/(dx) = lim_(deltaxrarr0) (deltay)/(deltax)`

`(dy)/(dx) = lim_(deltaxrarr0) ((2x+deltax)(deltax))/(deltax)`

`(dy)/(dx) = lim_(deltaxrarr0) ((2x+deltax)`

`(dy)/(dx) = 2x+0`

`(dy)/(dx) = 2x`

*So the derivative of x^2 is 2x as proved from first principles.*

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