  ### Home > APCALC > Chapter 4 > Lesson 4.3.1 > Problem4-114

4-114.

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

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

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

$y=\sin\left(x−3\right)$ 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.

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

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?