How do you implicitly differentiate #7xy- 3 lny= 4/x#?

1 Answer
Dec 29, 2015

#dy/dx=-(7x^2y^2+4y)/(7x^3y-3x^2)#

Explanation:

Find the derivative of each part.

Use product rule:

#d/dx(7xy)=7y+7xy'#

Recall that #d/dx(lnu)=(u')/u# through the chain rule:

#d/dx(-3lny)=-(3y')/y#

Rewrite #4/x# as #4x^-1#.

#d/dx(4x^-1)=-4x^-2=-4/x^2#

Add all these derivatives to find the complete differentiated expression, then solve for #y'#.

#7y+7xy'-(3y')/y=-4/x^2#

Multiply everything by #x^2y# to clear fractions.

#7x^2y^2+7x^3y(y')-3x^2y'=-4y#

#y'(7x^3y-3x^2)=-7x^2y^2-4y#

#y'=-(7x^2y^2+4y)/(7x^3y-3x^2)=dy/dx#