Question

What is the implicit derivative of `5=y/(x-y)`?

Answer 1

`(dy)/(dx)=y/x`

Explanation:

Quotient Rule for Derivatives:
`d(u/v)/dx = (v*(du)/(dx)-u*(dv)/(dx))/(v^2)`

With `u = y` and `v=(x-y)`
`color(white)("XXX")(du)/(dx) = (dy)/(dx)`
and
`color(white)("XXX")(dv)/(dx) = (d(x-y))/(dx) = 1-(dy)/(dx)`

Therefore
`color(white)("XXX")(d(y/(x-y)))/(dx) = ((x-y)(dy)/(dx) - y(1-(dy)/(dx)))/((x-y)^2)`

`color(white)("XXXXXXXX")=(x(dy)/(dx)-y)/((x-y)^2)`

`(d(5))/(dx)=0`

So, since `5= y/(x-y)`

`rArrcolor(white)("XXX")(d(5))/(dx) = (d(y/(x-y)))/(dx)`

`rArrcolor(white)("XXX")0 = (x(dy)/(dx)-y)/((x-y)^2)`

`rArrcolor(white)("XXX")x(dy)/(dx)-y=0`

`rArrcolor(white)("XXX")(dy)/(dx) = y/x`