logarithmic differentiation

Right i think i know this one but want to see if im right or does it need more simplifying!

Q. Find an expression dy/dx for the equation (1+x)sin(^2) x/(1+4x)(1-x)^3 using logarithmic differentiation.

A. ln y = ln(1+x) + ln((sin x)^2) - ln(1+4x) - ln(1-3)^3

1/y dy/dx = 1/(1+x) + cos x/sin x - 4/(1+4x) + 3/(1-x)

I then replaced cos x/sin x with cot x and subbed in y
therefore

((1+x)sin(^2) x/(1+4x)(1-x)^3)*(1/y dy/dx = 1/(1+x) + cot x - 4/(1+4x) + 3/(1-x))

Is this correct or is there more to it?
Thanks

Sorry that answe bit isnt quite right didnt cut and paste it right heres what i meant:


dy/dx = ((1+x)sin(^2) x/(1+4x)(1-x)^3)*(1/(1+x) + cot x - 4/(1+4x) + 3/(1-x))