Question

Show that `bb("dy"/"dx" = y/x)` in the following, where a and p are constant:

x^py⁴ = (x + y)^p+4, p ∈ N

Answer 1

x^py⁴ = (x + y)^p+4 

Taking log

log(x^py⁴) = log(x + y)^p+4

logx^p + 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`