A) \[{{e}^{x}}-\sin x\]
B) \[-{{({{e}^{x}}+\cos x)}^{-2}}\]
C) \[-({{e}^{x}}-\sin x)\,\,{{({{e}^{x}}+\cos x)}^{-2}}\]
D) \[(\sin x-{{e}^{x}})\,\,{{(\cos x+{{e}^{x}})}^{-3}}\]
Correct Answer: D
Solution :
\[y={{e}^{x}}+\sin x\] \[\frac{dy}{dx}={{e}^{x}}+\cos x\Rightarrow \frac{dx}{dy}=\frac{1}{{{e}^{x}}+\cos x}\] Diff. w.r.t y \[\frac{{{d}^{2}}x}{d{{y}^{2}}}=\frac{-1[{{e}^{x}}-\sin x]}{{{({{e}^{x}}+\cos x)}^{2}}}\times \frac{dx}{dy}=\frac{(\sin x-{{e}^{x}})}{{{({{e}^{x}}+\cos x)}^{3}}}\]
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