A) \[2x\,[1+\tan \,(\log x)]+x{{\sec }^{2}}(\log x)\]
B) \[x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\]
C) \[2x\,[1+\tan \,(\log x)]+{{x}^{2}}\,\,{{\sec }^{2}}(\log x)\]
D) \[2x\,[1+\tan \,(\log x)]+{{\sec }^{2}}(\log x)\]
Correct Answer: A
Solution :
\[x=\exp \left\{ {{\tan }^{-1}}\left( \frac{y-{{x}^{2}}}{{{x}^{2}}} \right) \right\}\] Þ \[\log x={{\tan }^{-1}}\left( \frac{y-{{x}^{2}}}{{{x}^{2}}} \right)\] Þ \[\frac{y-{{x}^{2}}}{{{x}^{2}}}\,=\,\tan (\log x)\] Þ \[y={{x}^{2}}\tan (\log x)+{{x}^{2}}\] Þ \[\frac{dy}{dx}=2x\,.\,\tan (\log x)+{{x}^{2}}\,.\,\frac{{{\sec }^{2}}(\log x)}{x}+2x\] Þ \[\frac{dy}{dx}=2x\tan (\log x)+x{{\sec }^{2}}(\log x)+2x\] Þ \[\frac{dy}{dx}=2x[1+\tan (\log x)]+x{{\sec }^{2}}(\log x)\].You need to login to perform this action.
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