A) \[2{{e}^{x}}\left( \sin \,x+\cos \,x \right)\]
B) \[2{{e}^{x}}\left( \cos x-\sin x \right)\]
C) \[2{{e}^{x}}\left( \sin \,x-\cos \,x \right)\]
D) \[2{{e}^{x}}\,\cos x\]
Correct Answer: B
Solution :
we have, \[f\left( x \right)={{e}^{x}}\sin x\] \[\Rightarrow \,f'\left( x \right)={{e}^{x}}\cos \,x+{{e}^{x}}\sin \,x={{e}^{x}}\left( \cos \,x+\sin x \right)\] \[\left( x \right)={{e}^{x}}\left( \cos x-\sin x \right)+{{e}^{x}}\left( \cos x+\sin x \right)\] \[=2{{e}^{x}}\cos x\] \[\Rightarrow f'''\left( x \right)=2\left[ {{e}^{x}}\cos x-{{e}^{x}}\sin x \right]=2{{e}^{x}}\left[ \cos x-\sin \,x \right]\]
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