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Question
If y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`, then show that `"dy"/"dx" = sinx/(1 - 2y)`.
Solution
y = `sqrt(cosx + sqrt(cosx + sqrt(cosx + ... ∞)`
∴ y2 = `cos x + sqrt(cos x + sqrt(cos x + ... ∞)`
∴ y2 = cos x + y
Differentiating both sides w.r.t. x, we get
`2y"dy"/"dx" = -sin x + "dy"/"dx"`
∴ `(1 - 2y)"dy"/"dx"` = sinx
∴ `"dy"/"dx" = sinx/(1 - 2y)`.
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