Calculus: More Calculus of Inverse Trig Functions

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Elementary Functions and Their Inverses
Clalculus of Inverse Trig Functions
More Calculus of Inverse Trig Functions Page [1 of 2]
All right, so now that we have these formulas at our disposal and we sort of add them to our arsenal of little facts that we know and understand, we can actually combine them with the different techniques of differentiation, like the product rule, and the quotient rule, and the chain rule and so on. Let' me just illustrate this really fast with some quickie examples.
Suppose I have f(x) = arcsin x^4 - 1. And suppose I want to find the derivative? Well, what's involved in taking the derivative? Well, nothing, now that I know the formula for arcsin x. All I have to do now is apply the chain rule. If I apply the chain rule, I just think about it like this: this is arcsin blop, so what's the derivative of arcsin blop? It's . So to find the derivative, what I do is I say, "Okay, the derivative of the outside, arcsin blop, the derivative is ." So I just peeled off the arcsin, I took the derivative of that. Now that I've got to multiply that by the derivative of the inside and the derivative of that is just 4x^3. So I multiply this by 4x^3, and so I see what the answer is. The answer is the derivative = . That's the derivative of this function. Just apply the chain rule and the formula for the derivative of arcsin. So, not a big deal at all. Again, arcsin blop, the derivative is , and then multiply that by the derivative of the inside, which is just the 4x^3. And that comes on top. So no problem.
How about another one? How about g(x) - some people actually get, you know, they see g and they say, "Oh, wait, what do you do with g's?" It's just the name of the function. You can call it Mary(x) or Todd(x). In fact, let's do that. Let's call this one Todd(x). It's just a name. But here it is, it's arctan. It's sort of appropriate that Todd(x)= arctan 3x. And so, I want to find the derivative of Todd. So how do you find the derivative of Todd? Well, what I'm going to do is I'm going to use the arctan fact that the derivative of arctan x = . So, in this case, I see that Todd is arctan blop. So Todd has an inside and an outside. So what do I do? Well, I first take the derivative of arctan blop. That would be . So the first thing I write is Todd'(x) = . So that's taking the derivative of arctan blop. So I just peeled away the arctan part. Now I've got to multiply that by the derivative of the inside. And the derivative of Todd's insides is just 3. And so I see the answer is . Or if you wanted to write Todd's derivative a little nicer, you could write it as , if you actually want to square out both those things. So there's the derivative of this slightly more exotic function, but again, just an application of a chain rule, so not a big deal at all.
So, now there's other things we can do. In fact, if you search really hard, you can actually get integral formulas. So if we go on a search, you can find them. And all they are is writing things backwards. It's just doing things backwards, and let's see if we can find them. Oh, here they are! Anyway, these facts are nothing more than the formulas we already saw last time. In fact, this one we derived for ourselves, but writing it the other way. This says if you want to integrate this very threatening-looking function, , what's the integral of that? Who knows? Well, we know, because we saw it must be arcsin x + a constant. And why is that? Because we're already established the fact that the derivative of this is that. So if the derivative of this equals that, that means if you anti-differentiate this, you get back to here. So it's just the formula's written backwards. So here's the formula. There's a good one for you. You can look at that for a while. Here's another one. How about this? The integral of . Well that's arccos x + c. If you look at this one, the integral of , well, that integral is going to be arctan x + c. And if you integrate , you can write that as arccot x + c. If you integrate , you're going to get arccsc x + c. Don't forget the constant. And then if you want to integrate , well that's just arcsec x. So you can see these are formulas that just sort of follow immediately from the derivative formulas we saw.
Now you can use these things in action. Let me just do an example for you real fast. Let's do an integral . This integral, just moments ago, would have been impossible, because what technique could you possibly use? We don't even know that many techniques, so who knows what you do to integrate that? I don't know, but we have this new arsenal and we just go through the arsenal and say, "Well, bingo, look at that. That's just arctan x." So, in fact, that has to become now a familiar one for us, and we just have to say, "Oh that's arctan x + c." Okay, so that wasn't too bad.
Now, how about one that's a little bit more interesting? So let's see if we can sort of up the interesting level. How about the integral of ? Now what in the world does that equal? My first thinking is all the formulas that we saw so far, they all seem to have a lot of square root and stuff, but they don't seem to really have the 5 in it. I don't see the 5 occurring at all. So this is looking a little bit threatening. Well, let's just go through and see if any of these look like a possible match. Well, that's probably not a match, so I'm not going to use that formula. That one we toss out, that's no good. How about this one? Well, that doesn't look too bad, except there's no minus sign there, but maybe that would work. Maybe I should hold that off and see what happens there. What about this one? No, because here I have something - something x^2. Here I've got the x^2 up front, so that's no good. This one's definitely no good. In fact, maybe none of these are going to work. Maybe this one I can't even do. This one, that looks pretty close, but these are in the wrong order. This is no good. This one - now this one, actually, is pretty close, because I have a -1 on top, that's good. On the bottom I've got something minus and then the x^2. In fact, if it weren't for those stupid 5's, I'd be done. So what can I possibly do? Well, look if you've got two stupid 5's, factor them out. So whenever you see stupid 5's together, a great idea - and, in fact, this technique will work even if you have 7's. You can factor them out, too. So let's factor them out and see what happens. If I factor it out, what I would see is . And now I can actually break up this square root. The times that is the times the square root of this stuff. And so, in fact, this actually equals the integral of . Now , that's just a constant number. And so we know that we could just pull that out of the integral. So, in fact, that could come out in front. We could just pull that right outside and then what am I left with? I'm left with the integral of , and that's exactly this formula. So it's arccos x. So, in fact, this whole thing was just a pretty basic inverse trig function with a little constant multiple out in front. arccos x + c. So even integrals that don't exactly look like they can be applied with or use some of the inverse trig functions, after a little teeny bit of algebraic stuff, you can actually find the little good nugget inside of there.
Anyway, there's some nice little integration and differentiation examples, just trying to bring all these new functions into our realm of calculus. I'll see you at the next lecture.