Dropping the class and screaming for the hills, I didn’t
sign on for this! Learning methods and shortcuts to quickly compute difference quotients for progressively more
complicated functions to save our sanity when we actually try to apply derivatives to anything later. Learning about
what kinds of situations derivatives apply to so we have something to hold onto as we descend into the madness. Clicking the next button and trying to get through the videos as fast as possible and hoping something sinks in.
We discuss how we will tackle the next phase of calculus; the derivative!
Video Lecture
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Text and Additional Details
Now that we’ve learned limits in a variety of abstract (and not so abstract) settings, we will begin a deep dive into one of the most prolific applications of limits: the derivative. We will discuss the derivative in detail in coming segments; here we aim to give a brief overview of what we will be covering moving forward.
The derivative is, at its heart, a method to determine how a function is changing at a give point. We have actually already started this investigation when we discussed instantaneous rate of change during our limits section. The derivative is a more generalized version of this idea - typically still referred as ’instantaneous rate of change’ even when the setting is more abstract than the versions we have already seen. Nonetheless, this may seem like a weirdly specific thing to care about, how often does it really help to look at how quickly a function is changing at some value after all? It turns out, however, this piece of information can be used in all kinds of different and clever ways.
Before we dive into applications of derivatives though, we first need to know how to calculate them. Preferably, we need to be able to calculate derivative values quickly and easily, especially of progressively more complicated functions. To this end we will begin by spending a significant amount of time learning rules and techniques necessary to compute these values quickly and easily, even when confronted with challenging and complicated functions. Again it may seem like this should be easy after our segment in limits, but computing a difference quotient for ... that would involve ultimately trying to simplify something like this:
... Hard pass. That looks like a nightmare. To avoid doing that kind of insanity, the first half of our time in derivatives will be spent learning how to handle functions like this differently - rules to calculate what we want without going through that kind of madness.
Once we have learned how to handle these more challenging functions, we can move on to the various applications of derivatives. This will range from determining when functions are increasing or decreasing, to sketching detailed graphs of complicated functions, to various optimization problems to name a few. Here we will learn how to recognize where derivatives might be useful in everyday language, and how to model seemingly unrelated problems using the mathematics of calculus.
So, as we move forward, keep in mind that we will be covering a lot of mechanics; mostly to make our lives easier when trying to compute the necessary difference quotient results when we have much more difficult real-world applications involving derivatives.
1 : So we are going to start our exploration of derivatives by...