if y= 1+(a/x-a)+(bx/(x-a)(x-b))+(cx^2/(x-a)(x-b)(x-c)) than show that dy/dx=y/x[(a/a-x)+(b/b-x)+(c/c-x)].............

Dear pradeep,

y = 1 + a/(x-a) +bx/(x-a)(x-b) + cx²/(x-a)(x-b)(x-c)

breaking into different  functions

y = 1 +    f(x)₁  +       f(x)₂       +        f(x)₃       

f(x)₁=a/(x-a)    ,        f(x)₂ =bx/(x-a)(x-b)   &   f(x)₃ =cx² /(x-a)(x-b)(x-c)

now

          dy/dx= d/dx(1) + d/dx(f(x)₁) +d/dx(f(x)₂) +d/dx(f(x)₃) ....................................1

now separatly differentiatiating f(x)_{1 ,} f(x)₂  , f(x)₃  by taking log

  (taking log in both sides)

          d/dx[log(f(x)₁)]  =  d/dx(loga/x-a)

                d/dx(f(x)₁/f(x)₁ = d/dx(loga - log(x-a) )                 

                     d/dx(f(x)₁₎=(-1/x-a)(f(x)₁

                       d/dx(f(x)₁) =    -a/(x-a)²

now separatly differentiate each of these and put in eq 1 ,u will get the required result