Derivatives of Inverse Trigonometric Functions
I don’t play a lot of video games (does saying “video games” make you sound old yet?) anymore, but I like to think of learning new mathematical tools as “leveling up.” You’re not just gaining ability – you’re also gaining access to new stuff.
One of the things you can do now that you have implicit differentiation is to convert those menacing inverse trig functions into expressions that are easier to deal with. And, bonus, the proof is cute. Let’s walk through the book’s proof:
1) Stipulate that arcsin is differentiable everywhere within its bounds that we established back in the section on inverse trig functions. As the book says, this is reasonable because an inverse trig function looks like a trig function rotated 90 degrees and bounded so it stays one-to-one.
2) Imagine a function, y=arcsin(x), with y bounded at + and – pi/2.
3) sin(y) = x, from definition of arcsin
4) Take the function from (3) and d/dx both sides, resulting in cos(y)*(dy/dx) = 1
5) So, dy/dx = 1/cos(y)
6) recall that . thanks to Pythagoras
7) Substitute the info in (6) into (5)
Amazing, right? Maybe now you understand what I mean when I say a proof is “cute.” I like little proofs like this because they’re like magic tricks. You follow along with the steps, all of which make sense, then BAM! A rabbit comes out of the hat.
Of course, the nice part about math is you get to examine the hat afterward and, sure enough, discover the rabbit.
The book then gives us the proof of the derivative of arc tan, which is pretty simple after you get the proof of arcsin.
You may be wondering to yourself how trig functions can be related to functions using only exponents and ratios. Well, what you have to understand is that trig isn’t a separate universe. Trigonometry describes the relationships between sides of triangles. That’s it. It ain’t magic.
Plus, by looking at the derivative functions, you can see that the old rules still apply. In arcsin’, the function stops being real at x=-1 and x=1. It’s bounded the same as arcsin!
Similarly, arctan’ makes sense. Notice that it is only bounded as x gets very large or very small, causing the function to go to 0.
It’s worth pondering over these to confirm in your brain that you can go from inverse trig to stuff that appears to be more simple. I know when I first got here it was intimidating and seemed like magic. BUT, in math, the best thing for me was to see the connections between things that appear dissimilar.
And, as ever, make sure with implicit differentiation you work lots of problems. It’s very easy once you’re in the swing of it. So, make it part of your working knowledge.
Next stop: Derivatives of Log Functions (SERIOUSLY, DON’T BE AFRAID!)