Rates of Change in the Natural and Social Sciences
I’m not going to linger too long here, since I think we covered a lot of stuff way back here.
Understanding this section really is just a matter of using the tools you’ve learned so far. Remember, a derivative is simply how one thing changes with another. It doesn’t matter of its speed changing over time or population changing over food supply, or quantity of syrup changing over quantity of pancakes.
The good thing to get reinforced here is that derivatives are not always about time. I think people often have this idea that derivatives have to have SOME element of time because you’re measuring how things change. There is an abstract sense in which this is true. For example, if you’re measuring how cost of gold changes with respect to the value of a dollar. Yes, the ratio would cease to change if time ceased to move forward. However, logically and mathematically time isn’t relevant. We’re simply measuring how one thing changes with another.
Part of what might confuse you even in that example is that you’re aware that time may matter. Indeed, in a perfect equation of reality, billions of variables would probably matter (time, temperature, belligerence between nations, etc.). However, in this case, we’re strictly looking at how one thing (gold) changes with another thing (value of the dollar). Don’t go sticking time where it doesn’t belong.
Hopefully, this section also gives you a nice sense of the power of derivatives. By simply knowing the two variables and their derivative (that is, picking two things and knowing how they change with respect to each other), you can determine the state of a system at any point in its evolution.
This is probably a section where it’d be good to work some problems. Slipping the derivative into your equation can sometimes feel a little odd, but after a while it’ll make perfect sense.
Next stop: Exponential Growth and Decay