In order to differentiate a function of a function, say y, =f(g(x)), where we have to find (dy)/(dx), we need to do (a) substitute u=g(x), which gives us y=f(u). Then we need to use a formula called Chain Rule, which states that (dy)/(dx)=(dy)/(du)xx(du)/(dx). In fact if we have something like y=f(g(h(x))), we can have (dy)/(dx)=(dy)/(df)xx(df)/(dg)xx(dg)/(dh)
We also use here quotient rule that is if f(x)=(g(x))/(h(x))
then (df)/(dx)=((dg)/(dx)xxh(x)-(dh)/(dx)xxg(x))/(h(x))^2
as g(x)=sin^2 3x, using chain rule (dg)/(dx)=2sin3x xx cos3x xx3
= 6sin3xcos3x and
as h(x)=cos(2x) then (dh)/(dx)=-sin2x xx 2=-2sin2x
Hence derivative of (sin^2 3x)/(cos2x) is
(6sin3xcos3x xx cos2x - (-2sin2x)xxsin^2 3x)/(cos^2 2x)
= (6sin3xcos3xcos2x+2sin2xsin^2 3x)/(cos^2 2x)
= (2sin3x(3cos3xcos2x+sin2xsin3x))/(cos^2 2x)
= (2sin3x(2cos3xcos2x+cos(3x-2x)))/(cos^2 2x)
= (2sin3x(2cos3xcos2x+cosx))/(cos^2 2x)
