Differentiate each of the following (i) `(x+5)^(7)` , (ii) `(x-1/x)`

Explanation

(i) Let `y = (x+5)^(7)`
Let `deltay` be an increment in y, correponding to an increment `deltax` in x.
Then, `y + deltay = (x+deltax + 5)^(7)`
`rArr delta y = (x+deltax + 5)^(7) - (x+5)^(7)`
`rArr(deltay)/(deltax) = ((x+deltax+5)^(7) - (x+5)^(7))/(deltax)`
`rArr (deltay)/(dx) = underset(deltaxrarr0)("lim")(deltay)/(deltax)`
`= underset(deltararrx)("lim") ((x+deltax+5)^(7) - (x+5)^(7))/(deltax)`
`= underset((x+deltax + 5) rarr (x+5)) ("lim") ([(x+deltax +5)^(7) - (x+5)^(7)])/([(x+deltax+5)-(x+5)])`
`= underset(urarra)("lim") ((u^(7) - a^(7))/(u-a))`, where `(x+deltax + 5) = u` and `(x + 5) = a`
`= 7a^((7-1))= 7(x+5)^(6), [ :' underset(urarra)("lim")((u^(n)- a^(n))/(u-a)) = na^(n-1)]`.
Hence, `(d)/(dx)(x+5)^(7) = 7(x+5)^(6)`.
(ii) Let `y = (x-1/x)`
Let `deltay` be an increment in y, corresponding to an increment `deltax` in x.
Then, ` y + deltax = [(x+deltax) - (1)/((x+deltax))]`
`rArr deltay = [(x+deltax) - (1)/((x+deltax))] - (x - 1/x)`
`rArry = deltax + {1/x- (1)/((x+deltax))]`
`rArr(deltay)/(deltax) = (delta + {((x+deltax)-x)/(x(x+delta))})/(deltax)`
`rArr (dy)/(dx) = underset(deltax rarr 0)("lim") (deltay)/(deltax) = underset(deltax rarr0)("lim") ({deltax+(deltax)/(x(x+deltax))})/(deltax)`
`= underset(deltax rarr0)("lim") [1+1/(x(x+deltax))]= (1+1/(x^(2)))`
Hence, `d/(dx) (x - 1/x) = (1+1/(x^(2)))`