Calculus: Derivatives of Hyperbolic Functions

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Elementary Functions and Their Inverses
The Hyperbolic Functions
Derivatives of Hyperbolic Functions Page [1 of 2]
Okay, so once you have new functions, it’s always great to ask, “What can be said with respect to calculus? Can you take the derivative of these functions?” We saw the graphs and they look extremely smooth, very smooth. And so it looks like they should have slopes of tangent lines, and so what are the derivatives?
Well, actually, that is not a hard question, because remember, for example, let’s just think of the hyperbolic sine. The hyperbolic sine is defined to be just a whole bunch of exponential functions. So, in fact, we could just take the derivative of this, take the derivative of that and that’s going to give you the derivative of the hyperbolic sine. Similarly, if you take a look at the definition of hyperbolic cosine, it’s just a whole bunch of exponential functions. We can take the derivative of that, and then we can easily figure out what the derivative of the hyperbolic cosine is. For example, let’s just take a look at hyperbolic sine for a second together.
If we take the derivative of that, what do we see? The derivative of the hyperbolic sine equals – well, what I’ll do is I'm going to peel off that ½ out in front. That’s just a constant, and then what’s the derivative of ex? The derivative of ex is, of course, itself, ex, minus – and then I’ve got to use a little teeny chain rule, because I have eblop. The derivative of eblop is eblop, so that’s eblop, peeled off the e, and now I've got to take the derivative of –x, and that contributes a –1. So I have to put a –1 multiple in front, which changes this minus to a plus. And so there’s the answer. The answer is . So that’s the derivative of the hyperbolic sine. Familiar? Absolutely! That’s precisely the hyperbolic cosine. So, just a with the trig functions, the derivative of hyperbolic sine turns out to be hyperbolic cosine. And so there is the first identity that you can actually figure out. And we just did on our own. The derivative of the hyperbolic sine equals the hyperbolic cosine.
And now, you can use the same little very simple idea to differentiate all the functions. And if you do that? What would you see? Well, I’ll just show you what you’d see. You can try them on your own if you want. The derivative of the hyperbolic cosine is actually hyperbolic sine. Notice it’s a little different than the good old-fashioned cosine. The regular cosine function, the trig function cosine, has a derivative of –sine. But in the hyperbolic case, it’s a little teeny bit different. The derivative of hyperbolic cosine is actually just old-fashioned hyperbolic sine with nothing in front. A little different.
What about hyperbolic tangent? It’s, not too surprising, hyperbolic secant. What about hyperbolic cotangent minus hyperbolic secant2 x, just like the classically cases. And here we run down these other ones: hyperbolic cosecant = -hyperbolic cosecant × hyperbolic cotangent. And here’s another one for you, the derivative of the hyperbolic secant = -hyperbolic secant × hyperbolic tangent. So there you go, and you can see now, it’s really clear that there are these sort of vague similarities. Maybe we’re off a little teeny bit, but at least the structure is all preserved and they sort of do look like trig functions. And that’s why they have these trig names. So even though they're not exactly always right on the mark, like in this case, the reality is they’re pretty darn close. And, as we’ve seen, in fact, these things really do occur in nature. And mathematicians love them, by the way. They’re great. At mathematics parties, they bring them out as party favors and people go nuts.
Now, let’s take some derivatives just to see these things in action. For example, suppose we have f(x) = e, so there’s an e, but I’m going to compound that by raising e to the sinh x. Wow! That looks like a bit one, but really not a big deal at all. All we have to do is apply the chain rule and use the fact that we know that the derivative of the hyperbolic sine is equal to the hyperbolic cosine. So you're going to see now this is really not as gripping and exciting as you might have first thought, because I’m going to use the chain rule. I’ve got eblop. What’s the derivative of eblop? It’s just eblop. So eblop, in this, the blop is hyperbolic sine. And I’ve got to multiply that whole thing by the derivative of the blop, which was saw is hyperbolic cosine. And so this derivative is just cosh x times esinh x. So again, just an application of the chain rule and this formula. Not a big deal.
How about a more exotic one? I’ll try to see if I can stump the band, but it’s sort of hard here. How about this one? g – this could be anything. This could be Tom, it could be Paul, it could be anything you want, but I’ll keep it g. ln cosh x2. Now here we’ve got to apply the chain rule a couple of times, because there’s a lot of nesting. Let’s see all the nesting. Well, there’s ln blop. So there’s one chain rule right there. And once I peel off that, I’m left with this. But that actually also has an inside. It’s cosh blop, and the blop is x2. So this is going to require two applications of the chain rule. So let’s see that now in action. We’d see g’(x) equals – okay, what’s the derivative of ln blop? It’s just 1/blop. So we have 1/cosh x2. That now has peeled off the natural log. Now I’m left with taking the derivative of the inside, which is this. But that actually has an inside and an outside. So, first, let’s take the derivative of cosh blop. What’s that? Well, I’ll remind you that the derivative of cosh blop = sinh blop. So, in fact, the derivative of cosh is just sinh blop, and then I peel that off and now I’ve got to take the derivative of the inside-inside, which is the derivative of x2, which I worked out in advance to be 2x. So the derivative of this function turns out to be × sinh x2 × 2x. Now, in fact, you can actually combine these things and notice that actually has a name. That’s called hyperbolic tangent. So you could actually combine these two things and write it as tanh x2 × 2x, if you want, or you can keep it like this. It’s sort of an option that you have.
Anywhere, there’s a whole bunch of stuff about hyperbolic functions. They're sort of fun, they're sort of groovy, they're sort of hip and they're sort of now. So enjoy them. I’ll see you at the next lecture.