A) \[7{{\cos }^{5}}x+35{{\cos }^{3}}x\]
B) \[7{{\cos }^{5}}x+35{{\cos }^{2}}x\]
C) \[-7{{\cos }^{5}}x+35{{\cos }^{2}}x\]
D) None of the above
Correct Answer: D
Solution :
We have,\[y=\sin x\] \[\Rightarrow \] \[\frac{dy}{dx}=\cos x\] Now, \[\frac{{{d}^{2}}}{d{{y}^{2}}}({{\cos }^{7}}x)\] \[=\frac{d}{dy}\left[ \frac{d}{dy}({{\cos }^{7}}x) \right]\] \[=\frac{d}{dy}\left[ 7{{\cos }^{6}}x(-\sin x)\frac{dx}{dy} \right]\] \[=\frac{d}{dy}\left[ -7{{\cos }^{6}}x\sin x\cdot \frac{1}{\cos x} \right]\left( \because \,\,\frac{dx}{dy}=\frac{1}{\cos x} \right)\] \[=\frac{d}{dy}[-7{{\cos }^{5}}x\sin x]\] \[=-7[-5{{\cos }^{4}}x\cdot \sin x\cdot \sin x+{{\cos }^{5}}x\cdot \cos x]\frac{dx}{dy}\] \[=-7[-5{{\cos }^{3}}x{{\sin }^{2}}x+{{\cos }^{5}}x]\left( \because \frac{dx}{dy}=\frac{1}{\cos x} \right)\] \[=35{{\cos }^{3}}x{{\sin }^{2}}x-7{{\cos }^{5}}x\] \[=-7{{\cos }^{5}}x+35{{\cos }^{3}}x(1-{{\cos }^{2}}x)\] \[=-42{{\cos }^{5}}x+35{{\cos }^{3}}x\]
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