Question

How do you implicitly differentiate `9=-y/x+xsiny`?

Answer 1

`dy/dx=(x^2siny+y)/(x-x^3cosy)`

Explanation:

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

`"differentiate " y/x" using the "color(blue)"quotient rule"`

`"differentiate " xsiny" using the "color(blue)"product rule"`

`rArr0=-((xdy/dx-y)/(x^2))+xcosy.dy/dx+siny`

`rArr(xdy/dx-y)/x^2=xcosy.dy/dx+siny`

.`rArrxdy/dx-y=x^2(xcosy.dy/dx+siny)`

`rArrxdy/dx-y=x^3cosy.dy/dx+x^2siny`

`rArrxdy/dx-x^3cosydy/dx=x^2siny+y`

`rArrdy/dx(x-x^3cosy)=x^2siny+y`

`rArrdy/dx=(x^2siny+y)/(x-x^3cosy)`