Differentiate :
(i) `(ax + b)^(m)` , (ii) `(3x+5)^(6)` , (iii) `sqrt(ax^(2) + 2bx + c)`

Explanation

(i) Let `y = (ax + b)^(m)`.
Put `(ax + b) = t`, so that `y = t^(m)` and `t = (ax + b)`.
`:. (dy)/(dt) = mt^(m-1)` and `(dt)/(dx) = a`.
So, `(dy)/(dx) = ((dy)/(dt) xx (dt)/(dx))`
`= amt^(m-1) = am (ax +b)^(m-1) [ :' t = (ax+b)]`.
(ii) Let `y = (3x+5)^(6)`.
Put `(3x+5) =t`, so that `y = t^(6)` and `t = (3x+5)`.
`:. (dy)/(dt) = 6t^(5)` and `(dt)/(dx) = 3`.
So, `(dy)/(dx) = ((dy)/(dt) xx (dt)/(dx))`
`= 18t^(5) = 18 (3x+5)^(5), [ :' t = (3x+5)]` .
(iii) Let `y = sqrt(ax^(2) + 2bx + c)`.
Put `(ax^(2) + 2bx + c) = t`, so that `y = sqrt(t)` and `t = (ax^(2) + 2bx + c)`.
`:. (dy)/(dx) = 1/2 t^(-1//2) = (1)/(2sqrt(t))` and `(dt)/(dx) = (2ax+2b)`
So, `(dy)/(dx) = ((dy)/(dt) xx (dt)/(dx)) = 1/(2sqrt(t)) xx 2(ax+b)`
`= ((ax + b))/(sqrt(t)) = ((ax+b))/(sqrt(ax^(2) + 2bx + c))`.