Find the derivative of each of the following from the first principle.
(i) `x^(2) + 3x+5` ,(ii) `x^(3) + 4x^(2) + 3x + 2`

Explanation

(i) Let `y = x^(3) + 3x + 5`.
Let `deltay` be an increment in y, corresponding to an increment `deltax` in x.
Then , `y + deltay = (x+ deltax)^(2) + 3(x+deltax) + 5`
`rArr deltay = [(x+deltax)^(2) + 3(x+deltax) + 5] - (x^(2) + 3x + 5)`
`rArr (deltay)/(deltax) = ([(x+deltax)^(2) + 3(x+deltax) + 5] - (x^(2) + 3x + 5))/(deltax)`
`rArr (dy)/(dx) = underset(deltax rarr 0)("lim") (deltay)/(deltax)`
`= underset(deltax rarr 0) ("lim") ([(x+deltax)^(2) + 3(x+deltax) + 5] - (x^(2)+ 3x + 5))/(deltax)`
`= underset(delta x rarr )("lim")((deltax)^(2) + 2x.deltax + 3deltax)/(deltax) = underset(deltax rarr 0)("lim") [(deltax) + 2x + 3] = (2x+3)`
`:. (d)/(dx) (x^(2) + 3x + 5) = (2x+3)`.
(ii) Let `y = x^(3) + 4x^(2) + 3x + 2`.
Let `delta y` an increment in y, correspondin to an increment `deltax` in x.
Then, `y + deltay = (x+deltax)^(3) +4(x+deltax)^(2) + 3(x+deltax)+2`
`rArr (deltay)/(deltax) = ([(x+deltax)^(3) + 4(x+deltax)^(2) + 3(x+deltax) + 2] - (x^(3) + 4x^(2) + 3x +2))/(deltax)`
`rArr (dy)/(dx) = underset(deltax rarr 0)("lim") (deltay)/(deltax)`
`[(x+deltax)^(3) + 4(x+deltax)^(2) + 3(x+deltax) + 2]`
`= underset(deltax rarr 0)("lim")(-(x^(3) + 4x^(2) + x + 3x+ 2))/(deltax)`
`= underset(deltaxrarr0)("lim")((deltax)^(3) + 3x.deltax(x+deltax)+4(deltax)^(2) + 8x . deltax+3deltax)/(deltax)`
`= underset(deltaxrarr0)("lim")[(deltax)^(2) + 3x^(2) + 3x.deltax + 4(deltax) + 8x+3]= (3x^(2) + 8x + 3)`
`:. (d)/(dx) (x^(3) + 4x^(2) + 3x + 2) = 3x^(2) + 8x + 3`.