Question

What is the implicit derivative of `4= xyln(xy)`?

Answer 1

`dy/dx = -y/x`

Explanation:

`4=xyln(xy)`
`:. 4=xy(ln x + lny)`

Applying the triple product rule we get:

`:. 0=(x)(y)(d/dx(ln x + lny)) + (x)(d/dxy)(ln x + lny) + (d/dxx)(y)(ln x + lny)`
`:. 0=xy(d/dxln x + d/dylnydy/dx) + x(d/dyydy/dx)(ln x + lny) + (1)(y)(ln x + lny)`
`:. 0=xy(1/x + 1/ydy/dx) + x(dy/dx)(ln x + lny) + (y)(ln x + lny)`
`:. 0=y + xdy/dx + x(dy/dx)(ln xy) + (y)(ln xy)`
`:. xdy/dx + xdy/dxln xy = -y - yln xy`
`:. xdy/dx(1+ln xy) = -y(1+ln xy)`
`:. xdy/dx = -y`
`:. dy/dx = -y/x`