what is the derivative of function y=sin^2xtan^4x/(x^2+1)^2

Question

what is the derivative function if you use logarithmic differentiation

Answer 1

Given function y = Sin^2x+tan^4x/(x^2+1)^2

Apply ln to each side.

ln y = ln Sin^2x+ln [Tan^4x/(x^2+1)^2]

ln y = 2 ln Sinx+ ln (Tan^4x)-ln(x^2+1)^2

ln y = 2 ln Sinx+4 lnTanx-2 ln (x^2+1)

Apply derivative to each side with respect of x.

dy/dx(ln y) = dy/dx[2 ln Sinx+4 lnTanx-2 ln (x^2+1)]

1/y dy/dx = 2dy/dx(ln Sinx)+4dy/dx(ln Tanx)-2 dy/dx(ln(x^2+1))

1/y dy/dx = 2(Cosx)1/Sinx+4(Sec^2x)1/Tanx-4x(1/(x^2+1)

1/y dy/dx = 2Cosx/Sinx+4Sec^2x/Tanx-4x/(x^2+1)

1/y dy/dx = 2Cotx+4Sec^2x/Tanx-4x/(x^2+1)

dy/dx = y[2Cotx+4Sec^2x/Tanx-4x/(x^2+1)]