We use here Chain Rule . 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)#
Here we have #y=f(x)=sin^2(2x+3)=(g(x))^2#,
where #g(x)=sin(h(x))# and #h(x)=2x+3#
Hence #(dy)/(dx)=(dy)/(dg)xx(dg)/(dh)xx(dh)/(dx)#
= #2g(x)xxcos(h(x))xx2#
= #2sin(2x+3)xxcos(2x+3)xx2#
an using #2sinAcosA=sin2A#, wecan simplify it as
#2sin(4x+6)#
