Question

What is the derivative of the function `y=sin(xy)`?

What is the derivative of the function `y=sin(xy)`?

Answer 1

`dy/dx = (ycos(xy))/(1-xcos(xy))`

Explanation:

Using implicit differentiation, the product rule, and the chain rule, we get

`d/dxy = d/dxsin(xy)`

`=> dy/dx = cos(xy)(d/dx(xy))`

`=cos(xy)[x(d/dxy)+y(d/dxx)]`

`=cos(xy)(xdy/dx + y)`

`=xcos(xy)dy/dx + ycos(xy)`

`=> dy/dx - xcos(xy)dy/dx = ycos(xy)`

`=> dy/dx(1-xcos(xy)) = ycos(xy)`

`:. dy/dx = (ycos(xy))/(1-xcos(xy))`