Question

Let f^(n) (a) denote the n-th derivative at a. If f(x)=sin(x)cos(2x)+cos(x)sin(2x) find f^(52) (π/18).

Answer 1

f(x)=sin 3x
f'(x)=3cos 3x=\[3\sin \left( \frac{ \pi }{ 2 }+3x \right)\]
\[f"(x)=3^{2}\cos \left( \frac{ \pi }{2 }+3x \right)=3^{2}\sin \left( \frac{ \pi }{ 2 } +\frac{ \pi }{2 }+3x\right)=3^{2}\sin \left( 2*\frac{ \pi }{2 }+3x \right)\]
\[f \prime \prime \prime \left( x \right)=3^{3}\cos \left( 2*\frac{ \pi }{2 } +3x\right)=3^{3}\sin \left( \frac{ \pi }{2 }+2*\frac{ \pi }{2 }+3x \right)=3^{3}\sin \left( \frac{ 3*\pi }{2}+3x \right)\]
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\[f ^{n}\left( x \right)=3^{n}\sin \left( n*\frac{ \pi }{2 }+3x \right)\]
put n=52,
\[x=\frac{ \pi }{18}and get the result.\]

Answer 2

thank you!