Question

If Z = 2x-y/ x+y

Find dz/dx and dz/dy

Answer 1

if $z={2x-y \over x+y}$

Find ${dz \over dx}$ and ${dz\over dy}$

let u= 2x-y and v=x+y

using quotient rule${dz \over dx}={v{du \over dx}-u{dv \over dx} \over v^2}$

${du \over dx}=2$

${dv \over dz}=1$

${dz \over dx}={(x+y)(2)-(2x-y)1 \over (x+y)^2}$

${dz \over dx}={2x +2y-2x+y \over (x+y)^2}$

${dz \over dx}={3y\over (x+y)^2}$

using quotient rule${dz \over dy}={v{du \over dy}-u{dv \over dy} \over v^2}$

${du \over dx}=-1$

${dv \over dz}=1$

${dz \over dx}={-1(x+y)-(2x-y)1\over (x+y)^2}$

${dz \over dx}={-x-y-2x+y \over (x+y)^2}$

${dz \over dx}={-3x \over (x+y)^2}$