How do you implicitly differentiate #4=y-(xe^y)/(2y-x)#?

1 Answer
Mar 31, 2017

# dy/dx=(4-e^y-y)/(xe^y+x+8-4y).#

Explanation:

To avoid Quotient Rule for Diffn., we first, rewrite the given eqn. as,

#xe^y=(2y-x)(y-4)=2y^2-xy-8y+4x, or,#

#xe^y+xy+8y=2y^2+4x=2(y^2+2x).#

#:. d/dx(xe^y+xy+8y)=d/dx{2(y^2+2x)}=2d/dx(y^2+2x).#

Following the Usual Rules of Diffn., we get,

#:.d/dx(xe^y)+d/dx(xy)+8d/dx(y)=2{d/dx(y^2)+2d/dx(x)}.#

Here, in the L.H.S., by the Product Rule, we have,

# d/dx(xe^y)=xd/dx(e^y)+e^yd/dx(x),#

#=xe^yd/dx(y)+e^y..........[because," the Chain Rule],"#

#:. d/dx(xe^y)=xe^ydy/dx+e^y.#

Also, #d/dx(xy)=xd/dx(y)+yd/dx(x)=xdy/dx+y.#

In the R.H.S., we have, #d/dx(y^2)=2yd/dx(y)=2ydy/dx.#

Altogether, we get,

# xe^ydy/dx+e^y+xdy/dx+y+8dy/dx=2{2ydy/dx+2}=4ydy/dx+4#

#:. (xe^y+x+8-4y)dy/dx=4-e^y-y,#

#rArr dy/dx=(4-e^y-y)/(xe^y+x+8-4y).#

Enloy Maths.!