What is the implicit derivative of #4= xyln(xy) #?
1 Answer
Nov 18, 2016
# 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 #
