Differentiation of functions containing sine and/or cosine is based on:
the derivative of $f\left(x\right)=sin\left(x\right)$ is $f\text{'}\left(x\right)=cos\left(x\right)$
the derivative of $f\left(x\right)=cos\left(x\right)$ is $f\text{'}\left(x\right)=\mathrm{-}sin\left(x\right)$
To find the derivative of a function containing sine and/or cosine you often need the other differentiation rules as well.
When differentiating a sinusoid, for instance, you need to take into account the chain rule and the rule of constants.