# Using first principles find the derivative of f(x) = sinx.

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`y = sinx ----(1)`

Let us say that when x changes by `deltax ` then y changes by `deltay` .

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

(2)-(1)

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

`deltay = 2cos(x+(deltax)/2)xxsin((deltax)/2)`

`deltay/deltax = (2cos(x+(deltax)/2)xxsin((deltax)/2))/(deltax)`

`deltay/deltax = (cos(x+(deltax)/2)xxsin((deltax)/2))/((deltax)/2)`

`dy/dx = lim_((deltax)rarr0)(cos(x+(deltax)/2)xxsin((deltax)/2))/(((deltax))/2)`

`dy/dx = lim_((deltax)/2rarr0)(cos(x+(deltax)/2)xxsin((deltax)/2))/((deltax)/2)`

`dy/dx = (lim_((deltax)/2rarr0)cos(x+(deltax)/2))xx(lim_((deltax)/2rarr0)(sin((deltax)/2))/((deltax)/2))`

We know that `lim_(xrarr0)sinx/x = 1`

`dy/dx = (cosx)xx1`

`dy/dx = cosx`

*So when f(x) = sinx then derivative of sinx is cosx.*

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