A) \[n\,\,{{\sin }^{n-1}}x\,\sin (n+1)x\]
B) \[n\,\,{{\sin }^{n-1}}x\,\cos (n-1)x\]
C) \[n\,\,{{\sin }^{n-1}}\,x\,\cos nx\]
D) \[n{{\sin }^{n-1}}x\,\cos (n+1)x\]
Correct Answer: D
Solution :
Given, \[y={{\sin }^{n}}x\,\cos nx\] \[\frac{dy}{dx}=n{{\sin }^{n-1}}x\cos x\,\cos nx-n{{\sin }^{n}}x\,\sin \,nx\] \[=n\,{{\sin }^{n-1}}x\,[\cos \,\,x\,\cos nx-\sin \,x\,\sin \,nx]\] \[=n\,\,{{\sin }^{n-1}}\,x\,\cos (n+1)x\]
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