find compelet derivative of y = ln(sinex))2 cos(tanx)

Question

differenciation

Answer 1

I read this as y=ln²(sin(x))cos(tan(x)).

Let p=ln(sin(x)), dp/dx=cos(x)/sin(x)=cot(x).

Let q=cos(tan(x)), dq/dx=-sin(tan(x))sec²(x).

d(p²)/dx=2pdp/dx=2ln(sin(x))cot(x).

y=p²q,

dy/dx=p²dq/dx+2pqdp/dx=

-ln²(sin(x))sin(tan(x))sec²(x)+2ln(sin(x))cos(tan(x))cot(x).