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Question
Show that `bb("dy"/"dx" = y/x)` in the following, where a and p are constant:
xpy4 = (x + y)p+4, p ∈ N
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Solution
xpy4 = (x + y)p+4
Taking log
log(xpy4) = log(x + y)p+4
logxp + logy4 = (p + 4) log(x + y)
p log x + 4 log y = (p + 4) log(x + y)
Differentiating both sides w.r.t. x, we get
`p."d"/"dx"logx + 4*"d"/"dx"logy = (p + 4)"d"/"dx"log(x + y)`
`p/x + 4(1)/y"dy"/"dx" = (p + 4)(1)/(x + y)(1 + "dy"/"dx")`
`"p"/x + 4/y"dy"/"dx" = ((p + 4))/((x + y)) + (p + 4)/((x + y))"dy"/"dx"`
`"dy"/"dx"[4/y - ((p + 4))/((x + y))] = (p + 4)/(x + y) - p/x`
`"dy"/"dx"[(4(x + y) -y(p + 4))/(y(x + y))] = (x(p + 4) -p(x + y))/(x(x + y)`
`"dy"/"dx"[(4x + 4y - py - 4y)/(y(x + y))] = (px + 4x - px - py)/(x(x + y)`
`"dy"/"dx"[(4x - py)/y] = (4x - py)/x`
`"dy"/"dx" = y/x`
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