Compare the graph of y = sin(x − 3) with its parent y = sinx.
y = sin(x − 3) looks just like y = sinx. 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 = sinx → y = sin(x+3)
Then y' = cosx → 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?