Implicit Differentiation and Related Rates

Let’s take a break from trying to work out how to differentiate everything in sight. It’s time to look at implicit differentiation, which is a nice generalization of regular differentiation. We’ll then see how to use this technique to solve word problems involving changing quantities. Knowing how fast one quantity is changing allows us to find how fast a different, but related, quantity is changing too. Anyway, the summary for this chapter is the same as the title:

• implicit differentiation; and

• related rates.

Consider the following two derivatives:

The first is just 2*x*, as we’ve seen. So isn’t the second one 2*y*? That would be ...

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