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Question
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.
Solution
For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is – 1.
Explanation:
Given that: `sqrt(x) + sqrt(y)` = 1
Differentiating both sides w.r.t. x
`1/(2sqrt(x)) + 1/(2sqrt(y)) * "dy"/"dx"` = 0
⇒ `1/sqrt(x) + 1/sqrt(y) "dy"/"dx"` = 0
⇒ `1/sqrt(y) "dy"/"dx" = (-1)/sqrt(x)`
⇒ `"dy"/"dx" = (-sqrt(y))/sqrt(x)`
∴ `"dy"/"dx"` at `(1/4, 1/4) = - sqrt(1/4)/sqrt(1/4)`
= – 1.
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