The Derivatives of the Sine and Cosine. [from Sensible Calculus]
(i) The Derivative of the Sine at $x=0$: $\sin'(0)=1$.
Check this visually with the mapping diagram CDD.DSMD0

CDD.DSMD0

(ii) The Derivative of the Cosine at $x=0$: $\cos'(0)=0$.
Check this visually with the mapping diagram CDD.DCMD0

CDD.DCMD0
Now that we have found the derivatives for sine and cosine at $x =0$, it is time to think of these derivatives more generally. In Theorem CCD.DSC we state these results.
Theorem CCD.DSC. i) $D_x\sin(x) = \sin'(x) =\cos(x)$.
$D_x\cos(x) = \cos'(x) = -\sin(x)$
Use the slider "Choose s(x):" to switch between the visulization for $s(x)=\sin(x)$ and $s(x)=\cos(x)$.