`1/(sqrt(x +2))`

Answer

`(-1)/(2(x+2)^(3/2))`

Explanation

`y=1/(sqrt(x+2))Rightarrow (y+deltay) = 1/(sqrt(x+deltax+2)) `
`Rightarrow (deltay)/(deltax)= {1/(sqrt(x+deltax+2))-1/(sqrt(x+2))}. 1/(deltax)`
`:. (dy)/(dx) = lim_(deltaxto0) (deltay)/(deltax)`
`= lim_(deltaxto0) 1/(deltax) {1/(sqrt(x+deltax+2))-1/(sqrt(x+2))}. 1/(deltax)`
`= (dy)/(dx) = lim_(deltax to0) (deltay)/(deltax)`
`= lim_(deltato 0) 1/(deltax) . ((sqrt(x+2)-sqrt(x+deltax+2))(sqrt(x+2)+sqrt(x+deltax+2)))/((sqrt(x+deltax+2))(sqrt(x+2))(sqrt(x+2)x sqrt(x+deltax+2)))`
`= lim_(deltaxto0) ({(x+2)-(x+deltax+2)})/((sqrt(x+deltax+2))(sqrt(x+2))). 1/((sqrt(x+2)+sqrt(x+deltax+2)))`
`= lim_(delta to0) (-1)/((sqrt(x+deltax +2)) (sqrt(x+2))). 1/((sqrt(x+2) + sqrt(x+deltax+2)))`
`= (-1)/(2(x+2)^(3//2))`
Hence, ` d/(dx) (1/(sqrt(x+2)))= (-1)/(2(x+2)^(3//2))`