I believe it is well established that (sin(x))' = cos(x).
If we let x = 2n, then substituting, we have sin(2n)' = cos(2n). However, the chain rule suggests otherwise: sin(2n)' = 2cos(2n).
In fact, graphically, we see that the chain rule is indeed correct, and the substitution of variables yields an incorrect answer.
What is wrong with this substitution that makes it a false statement?
\begin{eqnarray*} \frac{d}{dx} \sin (x) =\cos(x) \end{eqnarray*} When you substitute $x=2n$ you get \begin{eqnarray*} \frac{d}{d(\color{red}{2}n)} \sin (2n) =\cos(2n) \end{eqnarray*} Can you spot where the "extra" factor of $2$ comes from ?