9: Chain Rule and Inverse Trig Functions

”I am in the world only for the purpose of composing.” - Franz Schubert

Just as before, we have fully explored the relationship between derivatives and the four major operations, and forgotten about the fifth: composition. Chapter nine is an exploration on how differentiation interacts with composition.

First we see the nuts and bolts of the operation by explicitly stating the formula for the chain rule. After a few examples, the reader should feel competent enough to take the derivative of most combinations of functions they could imagine.

Of course, one cant mention composition without thinking about inverse functions again. We discover a formula for taking the derivative of an inverse function as a direct result of the chain rule.

Since inverse functions are so popular, we introduce the six new inverse trigonometric functions. These functions have specific restricted domains/ranges, but all ultimately are used to reverse one of the six standard trigonometric functions. We find all of their graphs, and learn how to evaluate them at points or in composition with other trigonometric functions.

Since these six functions are all continuous and smooth, they also have derivatives. We use our new inverse derivative formula to find out exactly what the derivative of each of our six new inverse trigonometric friends will be. As always, we get a little practice in using these new derivatives in combination with our old derivative shortcuts that we have collected along the way.

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