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Question
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.
Options
`cos/(2y - 1)`
`cosx/(1 - 2y)`
`sinx/(1 - 2y)`
`sinx/(2y - 1)`
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Solution
If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to `cos/(2y - 1)`.
Explanation:
Given that: y = `sqrt(sinx + y)`
Differentiating both sides w.r.t. x
`"dy"/"dx" = 1/(2sqrt(sinx + y)) * "d"/"dx" (sin x + y)`
⇒ `"dy"/"dx" = 1/(2sqrt(sinx + y)) * (cos x + "dy"/"dx")`
⇒ `"dy"/"dx" = 1/(2y) * [cos x + "dy"/"dx"]`
⇒ `"dy"/"dx" = cosx/(2y) + 1/(2y) * "dy"/"dx"`
⇒ `"dy"/"dx" - 1/(2y) * "dy"/"dx" = cosx/(2y)`
⇒ `(1 - 1/(2y))"dy"/"dx" = cosx/(2y)`
⇒ `((2y - 1)/(2y)) "dy"/"dx" = cosx/(2y)`
⇒ `"dy"/"dx" = cosx/(2y) xx (2y)/(2y - 1)`
⇒ `"dy"/"dx" = cosx/(2y - 1)`
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