### Home > CALC > Chapter 4 > Lesson 4.3.1 > Problem4-103

4-103.

Differentiate the following functions. Determine if the function is differentiable for all reals.

1. $y =\operatorname{sin}(x-3)$

Compare the graph of $y =\operatorname{sin}(x-3)$ with its parent $y =\operatorname{sin}x$.

$y =\operatorname{sin}(x-3)$ looks just like $y =\operatorname{sin}x$. 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.

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

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

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

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