A) \[2\text{ }sec\text{ }x{{(sec\text{ }x+tan\text{ }x)}^{2}}\]
B) \[2\text{ }sec\text{ }x{{(sec\text{ }x-tan\text{ }x)}^{2}}\]
C) \[2\,sec\,x\]
D) None of the above
Correct Answer: A
Solution :
\[y=\frac{\sec x+\tan x}{\sec x-\tan x}\] \[\Rightarrow \]\[y=\frac{\sec x+\tan x}{\sec x-\tan x}\times \frac{\sec x+\tan x}{\sec x+\tan x}\] \[\Rightarrow \]\[y=\frac{{{(\sec x+\tan x)}^{2}}}{{{\sec }^{2}}x-{{\tan }^{2}}x}\] \[\Rightarrow \]\[y=\frac{{{(\sec x+\tan x)}^{2}}}{1}\] On differentiating with respect to\[x,\] \[\frac{dy}{dx}=2(\sec x+\tan x)(\sec x\tan x+{{\sec }^{2}}x)\] \[=2(\sec x+\tan x)\sec x(\tan x+\sec x)\] \[=2\sec x{{(\sec x+\tan x)}^{2}}\]
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