Question

Help with calculus:

Find the tenth (!) derivative of y, if
y = (sinx)*sin(2x)*(sin(3x))

Answer 1

first use product to sum formulas

Answer 2

\[\sin x \times \sin 2x =\frac{ -1 }{2 }( \cos(x+2x) +\cos(x-2x))=\frac{ -1 }{2 }( \cos(3x) +\cos(x))=\]
\[\frac{ -1 }{2 }( \cos(3x) +\cos(x)) \sin 3x =\frac{ -1 }{4 }\sin (6x) +\frac{ -1 }{2 }\sin 3x \cos x\]

Answer 3

again use product to sum formulas
\[\frac{ -1 }{4 }\sin 6x +\frac{ -1 }{4 }(\sin(3x+x)+\sin(3x-x))\]

Answer 4

\[\frac{ -1 }{4 }(\sin 6x +\sin 4x +\sin 2x)\]

Answer 5

\[y=\sin(ax) \rightarrow y^{(n)}=a^{n} \sin (ax+n \frac{ \pi }{2 })\]