Question

differentiate 2y=x^4 sin(tan(x^3y))

Answer 1

2y=x^4 sin(tan(x^3y))
take log both side.

log(2y) = 4 sin(tan(x^3y)) *log(x)
now differentiate both side w.r.t x
d(log(2y))/dx = d(4 sin(tan(x^3y)) *log(x))/dx
apply the chain rule of differentiation.

(2/y)*y' = 4 cos(tan(x^3y)) *(sec(x^3y))^2 * 3y(x^(3y-1) *y' *log(x) + 4 sin(tan(x^3y)) *1/x

we can also write upper eqn in following way.

(2/y)*y' - 4 cos(tan(x^3y)) *(sec(x^3y))^2 * 3y(x^(3y-1) *y' *log(x) = 4 sin(tan(x^3y)) *1/x
now take y' as common from L.H.S term. and proceed it.
y' = 4 sin(tan(x^3y)) *(1/x) /(2/y) - 4 cos(tan(x^3y)) *(sec(x^3y))^2 * 3y(x^(3y-1) *log(x).