Math 221 Derivative Functions Discussion

This discussion builds your understanding of how to use the limit definition of derivative to find one function’s derivative as another function. Our textbook introduces the basic ideas in “Derivative Functions” within section 3.2. This discussion also touches on the relationship between differentiability and continuity, as discussed in “Derivatives and Continuity” in section 3.2.

Part 1

Use the limit definition of derivative to find derivative functions for each of the following. If you work out the complete derivatives, you will probably want to do it on paper before posting, and that paper work is likely to be the main value of this exercise. But remember that you can use this discussion to talk about questions you run into, thoughts you have along the way, etc., as well as about complete answers you find.

\[f(t) = \frac{1 }{2 }t^2 + t\]

\[f(s) = \sqrt{s^2-1}\]

Part 2

Based on their derivatives, where are the functions in Part 1 continuous? Is what you infer from the derivatives consistent with what you would expect by inspecting the functions?