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

4-103.

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

*y*= sin(*x*− 3)

Compare the graph of *y* = sin(*x* − 3) with its parent *y* = sin*x*.

*y* = sin(*x* − 3) 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(*x*+3)

Then *y*' = cos*x* → *y*' = cos(*x* + 3)

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?