Question

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

Answer 1

`=>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)`

Answer 2

`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)`