Differentiate the following functions. Determine if each function is differentiable for all real values of $x$.

1. $y=\sin\left(x-3\right)$

   Hint (a):

   Compare the graph of $y=\sin\left(x−3\right)$ with its parent $y=\sin x$.

   Hint (a):

   $y=\sin\left(x−3\right)$ looks just like $y=\sin x$. Their periods are the same. Their amplitudes are the same. Their maximum and minimum $y$-values are the same. Their SLOPES are the same. The only difference is their horizontal locations.

   Answer (a):

   The slopes will shift with the graph:
   If $y=\sin x→y=\sin\left(x+3\right)$
   Then $y^\prime=\cos x→y^\prime=\cos\left(x+3\right)$

2. $f ( x ) = \left\{ \begin{array} { c c } { 4 - x ^ { 2 } } & { \text { for } x < 1 } \\ { ( x - 1 ) ^ { 3 } + 3 } & { \text { for } x \geq 1 } \end{array} \right.$

   Hint (b):

   The derivative will be a piecewise function as well. Differentiate each piece separately.

   Hint (b):

   Consider the boundary point of the derivative. Examine the two pieces at $x = 1$. Is the derivative continuous?