let y=\[sin(ax+b)\]
\[y_1=acos(ax+b)=asin(ax+b+ \frac{\Pi }{2})\]
\[y_2=a^2cos(ax+b+ \frac{\Pi }{2})=a^2sin(ax+b+\frac{2\Pi }{2})\]
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\[y_n=a^nsin(ax+b+\frac{n\Pi }{2})\]
Boundary condition: n must be integer
problem1.Find the 3rd derivative of \[sin2x cos2x\] at x=0
solution
sin2x cos2x = \[\frac{2sin2xcos2x}{2}= \frac{sin4x}{2}\]
here n=3 a=4 b=0 x=0
\[y_3=4^3sin(4x+0+\frac{3\Pi }{2})\]=-64
