If `y=1+alpha/((1/x-alpha))+(beta//x)/((1/x-1)(1/x-beta))+(gamma//x^2)/((1/x-alpha)(1/x-beta)(1/x-gamma)),` find `(dy)/(dx)`

Explanation

`y=1+frac{alpha}{(frac{1}{x}-alpha)}+frac{beta / x}{(frac{1}{x}-alpha)(frac{1}{x}-beta)}+frac{gamma x^{2}}{(frac{1}{x}-alpha)(frac{1}{x}-beta)(frac{1}{x}-gamma)}`

`Rightarrow y=1+frac{a x}{1-a x}+frac{gamma x}{(1-a x)(1-beta x)}+frac{gamma x}{(1-alpha x)(1-beta x)(1-gamma)}`

Taking log on both sides, we get

`log y=log 1+log alpha x-log (1-alpha x)+log beta x-log (1-alpha x)-log (1-beta x)+log gamma x-log (1-alpha x)-log (1-beta x)-log (1-gamma x)`

`Rightarrow log y=log alpha x-3 log (1-alpha x)+log beta x-2 log (1-beta x)+log gamma x-log (1-gamma x)`

`Rightarrow frac{1}{y} frac{d y}{d x}=frac{1}{x}+frac{3 alpha}{1-alpha x}+frac{1}{x}+frac{2 beta}{1-beta x}+frac{1}{x}+frac{gamma}{1-gamma x}`

`Rightarrow frac{1}{y} frac{d y}{d x}=frac{1}{x}[3+frac{3 alpha}{(frac{1}{x}-alpha)}+frac{2 beta}{(frac{1}{x}-beta)}+frac{gamma}{(frac{1}{x}-gamma)}]`

`Rightarrow frac{d y}{d x}=frac{y}{x}[frac{alpha}{(frac{1}{x}-alpha)}+frac{beta}{(frac{1}{x}-beta)}+frac{gamma}{(frac{1}{x}-gamma)}]`