 With now the six hyperbolic inverse functions in hand, we can talk about their derivatives too. The derivative of inverse cinch, or arc cinch if you prefer, is going to be 1 over the square root of 1 plus x squared. Let's see how that is. If you want to compute the derivative of this, you can actually compute it implicitly because we know the derivative of cinch. Notice that if y equals cinch inverse of x, that implies that cinch of y is equal to x. We can take the derivative of both sides with respect to x. We then would see that on the left-hand side, you're going to get cos of y times by y prime. This is equaled, and on the right-hand side, you take the derivative of x, you're going to get a 1. So solving for cos, you're going to get that y prime equals 1 over cos of y. How do you deal with that relating the original expression we have over here? Notice that cinch of y equals x. Well, you can play around with the Pythagorean relationship for cinch and cos. If you take cos of y minus, sorry, cos squared of y minus cinch squared of y, this is equal to 1. Solving for cos, you're going to get cos of y is equal to the square root of 1 plus cinch of y, cinch squared of y, excuse me, and this is advantageous because cinch of y is equal to x. So you get cos of y is equal to square root of 1 plus x squared, for which we see that right here. So the derivative of the inverse of cinch is going to be 1 over the square root of 1 plus x squared. The derivative of arc cos, that is the inverse of cos, we're going to see right here by similar argument, you would see this is going to be 1 over the square root of x squared minus 1. There is sort of a difference of sign there that comes about because of the sign change between cinch and cos there. If we wanted to, we could calculate the derivative of the inverse of hyperbolic tangent, that's going to be 1 over 1 minus x squared. If you take the derivative of the inverse of hyperbolic cotangent, you likewise get the same formula, 1 over 1 minus x squared. Now you should notice that these formulas are identical, but these are not the same functions. And that's because of a difference of domains and ranges, particularly it's the domain that we should be talking about here. And we talked about this in a previous video when we introduced the hyperbolic inverse functions, while the domain for inverse cinch is going to be all real numbers, the domain of inverse cos is actually only going to be when x is greater than equal to 1. When it comes to hyperbolic tangent inverse, its domain is going to be from negative 1 to 1. On the other hand, the inverse of hyperbolic cotangent, this is when the absolute value is greater than 1. In other words, you need that x is less than negative 1 or x is greater than 1. So the domains of the inverse hyperbolic tangent and the inverse hyperbolic cotangent, they're exact opposite of each other, they're complementary in their domains right there. So the formula, the formula is going to be the same, but the domains are different, that's an important distinction. The derivative of the inverse hyperbolic secant function is going to be negative 1 over x times the square root of 1 minus x squared. And then the derivative of the inverse of hyperbolic cosecant is going to equal negative 1 over the absolute value of the square root of x squared plus 1. So formulas again are very similar, but again remember, domains here, the domain of the inverse hyperbolic secant function is going to be absolute value of x is greater than 1, just like the inverse hyperbolic cotangent. And that's basically because of the similarities of their functions there. And then lastly, the domain of the inverse of hyperbolic cosecant, that's everything except for zero. And so with those domain exceptions, we're going to get these formulas for their derivatives. So how do we calculate the derivative of a function involving a hyperbolic inverse? Well, just like we would in the other function, let's take for example here, take the derivative of inverse of hyperbolic tangent composed with sine. So we see two functions in play here, we have an inner function which is sine of x, we have this outer function, which is going to be the inverse of hyperbolic tangent. So by the chain rule, when we calculate the derivative of this thing, let's call the function here y. So y prime is going to equal, we take the outer derivative. So what was the derivative of hyperbolic inverse tangent again, you're going to get one over one minus something square at the inner function. So inside of that, we put the inner function sine of x, then we have to multiply that by the inner derivative, the derivative of sine is equal to cosine of x. And so we can put that together, try to simplify it perhaps, we get cosine of x on top, we're going to get one minus sine of x sine square on the bottom, for which I could simplify that one one minus sine square by the usual Pythagorean identity for circular trigonometry, one minus sine square is actually cosine squared. And so we get cosine of x on top, we could simplify that just to be one over cosine, which is the same thing as secant. So we actually have found a function whose derivative is equal to secant, which is kind of a fun little observation here. And that's going to then end our discussion here of hyperbolic functions, we talked about the hyperbolic functions, their inverses and the derivatives of all of these functions. If you learned anything in these videos, by all means, please hit the like button, subscribe to see more videos like this in the future. And always, if you have any questions, please post them in the comments, I'm glad to answer them. I'll see you next time, everyone in chapter four. Bye.