I am struggling a bit in computing the differential of the following function. In particular, I am confused by differentiating w.r.t. $\cos(x)$ and $\sin(x)$.
Let $f: S^1 \rightarrow \mathbb{R}^2$ given by $f(\cos(x), \sin(x)) = (\sin(2x)\cos(x), \sin(2x)\sin(x))$.
Using the comment of @Fakemistake. I have that the differential is $$df = \begin{pmatrix} 4sin(x)cos(x) & 2cos^2(x) \\ 2sin^2(x) &4sin(x)cos(x)\end{pmatrix}$$