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Question
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants: `log((x^20 - y^20)/(x^20 + y^20))` = 20
Solution
`log((x^20 - y^20)/(x^20 + y^20))` = 20
∴ `(x^20 - y^20)/(x^20 + y^20)` = e20 = k ...(Say)
∴ x20 – y20 = kx20 + ky20
∴ (1 + k)y20 = kx20 + ky20
∴ `y^20/x^20 = (1 - k)/(1 + k)`
∴ `y/x = ((1 - k)/(1 + k))^(1/20)`, a constant
Differentiating both sides w.r.t. x, we get
`"d"/"dx"(y/x)` = 0
∴ `(x"dy"/"dx" - y."d"/"dx"(x))/(x^2)` = 0
∴ `x"dy"/"dx" - y xx 1` = 0
∴ `x"dy"/"dx"` = y
∴ `"dy"/"dx" = y/x`.
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