Question

`d/dxsinx cosy`?

Answer 1

`cos(y)cos(x)`

Explanation:

We have
`d(sin(x)cos(y))/dx`
`cos(y)d(sin(x))/dx`
since `cos(y)=const.`
we get
`cos(y)cos(x)`

Answer 2

`d/dxsinxcosy=cosxcosy-sinxsinydy/dx`

Explanation:

We want to find `d/dxsinxcosy` assuming `y=y(x)`.

In this case, we use the product rule:

`d/dx uv=v(du)/dx+u(dv)/dx`

where `u=u(x),v=v(x)`

So

`d/dxsinxcosy=cosyd/dxsinx+sinxd/dxcosy`

We know `d/dxsinx=cosx`. To find `d/dxcosy`, we use the chain rule:

`d/dxf(g(x))=f^'(g(x))g^'(x)`

So

`d/dxcosy=-sinyd/dxy=-dy/dxsiny`

And

`d/dxsinxcosy=cosxcosy-dy/dxsinxsiny`