Chapter 1.3, Problem 78E

Use the approach of Exercise 77 to show that

$\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left(f\left(x\right)+c\right)$

For any costant $c$.[Hint: Compare the tangent lines to the graph of $f\left(x\right)$ and $f\left(x\right)+c$

]

Draw two graphs of your choice that represent a function $y=f\left(x\right)$ and its vertical shift $y=f\left(x\right)+3$

Pick a value of $x$ and consider the points $\left(x,f\left(x\right)\right)$ and $\left(x,f\left(x\right)+3\right)$. Draw the tangent lines to the curves at these points and describe what you observe about the tangent lines.

Based on your observation in part (b), explain why $\frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left(f\left(x\right)+3\right)$