Differentiate #sin y=xy^3+ylnx#?

2 Answers
Aug 30, 2017

#=>dy/dx=(xy^3+y)/(xcosy-3x^2y^2-xlnx)#

Explanation:

#siny=xy^3+ylnx#
Differentiating both sides with respect to #x# we get

#cosydy/dx=y^3+x×3y^2dy/dx+y/x+lnx×dy/dx#
#=>xcosydy/dx=xy^3+3x^2y^2dy/dx+y+xlnxdy/dx#
#=>dy/dx=(xy^3+y)/(xcosy-3x^2y^2-xlnx)#

Aug 30, 2017

#dy/dx=(xy^3+y)/((xcosy-3x^2y^2-xlnx)#

Explanation:

#color(blue)"differentiate implicitly with respect to x"#

#"differentiate "xy^3" and "ylnx" using the "color(blue)"product rule"#

#cosy.dy/dx=(x.3y^2dy/dx+y^3)+(y/x+lnx.dy/dx)#

#dy/dx(cosy-3xy^2-lnx)=y^3+y/x#

#rArrdy/dx=(y^3+y/x)/(cosy-3xy^2-lnx)#

#color(white)(rAeedy/dx)=(xy^3+y)/(xcosy-3x^2y^2-xlnx)#