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Question
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : `sin((x^3 - y^3)/(x^3 + y^3))` = a3
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Solution
`sin((x^3 - y^3)/(x^3 + y^3))` = a3
`(x^3 - y^3)/(x^3 + y^3)` = sina3 = b
`(x^3 - y^3)/(x^3 + y^3)` = b
x3 – y3 = b(x3 + y3)
x3 – y3 = bx3 + by3
x3 – bx3 = by3 + y3
x3(1 – b) = y3(b + 1)
`y^3/x^3 = (1 - b)/(1 + b)` = e
`y^3/x^3` = c .....(1)
y3 = cx3
Differentiating both sides w.r.t. x, we get
`3y^2"dy"/"dx"` = c.3x2
`(y^2dy)/(dx)` = cx2
`"dy"/"dx" c x^2/y^2`
`"dy"/"dx" = y^3/x^3. x^2/y^2` ....from(1)
`"dy"/"dx" = y/x`.
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