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Question
Show that `"dy"/"dx" = y/x` in the following, where a and p are constants : x7.y5 = (x + y)12
Solution
x7.y5 = (x + y)12
∴ (logx7.y5) = log(x + y)12
∴ logx7 + logy5 = log(x + y)12
∴ 7logx + 5logy = 12log(x + y)
Differentiating both sides w.r.t. x, we get
`7 xx (1)/x + 5 xx (1)/y."dy"/"dx" = 12 xx (1)/(x + y)."d"/"dx"(x + y)`
∴ `(7)/x + (5)/y."dy"/"dx" = (12)/(x + y).(1 + "dy"/"dx")`
∴ `(7)/x + (5)/y."y"/"dx" = (12)/(x + y) + (12)/(x + y)."dy"/"dx"`
∴ `((5)/y - 12/(x + y))"dy"/"dx" = (12)/(x + y) - (7)/x`
∴ `[(5x + 5y - 12y)/(y(x + y))]"dy"/"dx" = (12x - 7x - 7y)/(x(x + y)`
∴ `[(5x - 7y)/(y(x + y))]"dy"/"dx" = (5x - 7y)/(x(x + y)`
∴ `(1)/y."dy"/"dx" = (1)/x`
∴ `"dy"/"dx" = y/x`.
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