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Question
Choose the correct alternative.
If y = `sqrt("x" + 1/"x")`, then `"dy"/"dx" = ?`
Options
`("x"^2 - 1)/(2"x"^2sqrt("x"^2 + 1))`
`(1 - "x"^2)/(2"x"^2("x"^2 + 1))`
`("x"^2 - 1)/("2x"sqrt"x"sqrt("x"^2 + 1))`
`(1 - "x"^2)/("2x"sqrt"x"sqrt("x"^2 + 1))`
Solution
`("x"^2 - 1)/("2x"sqrt"x"sqrt("x"^2 + 1))`
Explanation:
y = `sqrt("x" + 1/"x")`
Differentiating both sides w.r.t.x, we get
`"dy"/"dx" = 1/(2sqrt("x" + 1/"x")) * "d"/"dx" ("x" + 1/"x")`
`= 1/(2sqrt(("x"^2 + 1)/"x")) * (1 - 1/"x"^2)`
`= sqrt"x"/(2sqrt("x"^2 + 1)) * (("x"^2 - 1)/"x"^2)`
`= ("x"^2 - 1)/("2x"sqrt"x"sqrt("x"^2 + 1))`
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