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When you have a composition of three or more functions, I find the "stuff" method particularly useful, because it's recursive, and you only have to focus on one function at a time -- the "outside" function -- the final function in the "chain".
We'll approach this topic via four examples, making good use of the chain rule:
And that's pretty much the graphical version of the Preview!:)
(WARNING: don't be fooled by the notation:
This is an interesting place to start, because the square function isn't actually invertible!
By convention, we restrict the domain, and choose to consider
the square function on only half of its domain (positive values of
And you can see what goes wrong if we try to reflect the part to the
left of the
It's unfortunate that we, being lazy mathematicians, will typically use
Hence
We simply assert that, if the two functions are equal, then their derivatives must be equal, as well:
So
Notice that it's our friend the hyperbola. Notice also
that this function is odd -- it should be the derivative of an even
function. We can extend the
Look at sine on this interval. If it is increasing everywhere, its inverse must be increasing everywhere: and what does that say about its derivative?
There are two other functions which deserve our attention. We call their inverses "arctan" and "arccos".
Do exactly the same thing that we've just done in these three examples, but with two other important functions: