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Question
Find the derivatives of the following:
`sqrt(x) = "e"^((x - y))`
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Solution
`sqrt(x) = "e"^((x - y))`
Squarting on both sides
`(sqrt(xy))^2 = ["e"^((x - y))]^2`
xy = `"e"^(2(x - y))` .......(1)
`x * ("d"y)/("d"x) + y * 1 = "e"^(2(x - y)) xx 2(1 - ("d"y)/("d"x))`
`x ("d"y)/("d"x) + y = 2"e"^(2(x - y)) - 2"e"^(2(x - y)) ("d"y)/("d"x)`
`x ("d"y)/("d"x) + 2"e"^(2(x - y)) ("d"y)/("d"x) = 2"e"^(2(x - y)) - y`
`[x + 2"e"^(2(x - y))] ("d"y)/("d"x) = 2"e"^(2(x - y)) - y`
`("d"y)/("d"x) = (2"e"^(2(x - y)) - y)/(x + 2"e"^(2(x - y))`
`("d"y)/("d"x) = (2xy - y)/(x + 2xy)`
By equation (1)
`("d"y)/("d"x) = (y(2x - 1))/(x(1 + 2y))`
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