 So we'll introduce a couple of new functions known as the hyperbolic functions. Now as a general rule, we can define new functions in terms of old functions, so I might define some function to be e raised to power sine of 3x. Generally it's not worth giving the new functions a new name, but sometimes it's convenient. And one of the times it's convenient is for what are known as the hyperbolic sine and cosine. And the hyperbolic sine and cosine are defined as hyperbolic sine, sometimes pronounced cinch, is e to the power x minus e to the minus x over 2, and the hyperbolic cosine sometimes referred to as cosh, e to the power x plus e to the power minus x over 2. Now a good rule in mathematics is mathematicians aren't very good at coming up with new names. And you can see that because these new functions are called the hyperbolic sine and cosine. But you might wonder why sine and cosine. So here's another good rule in mathematics. If a mathematician names something after something else, then in their minds at least there is some sort of connection. Now sometimes that connection is not readily apparent, but in this case there is at least a similarity between the hyperbolic sine and cosine and the regular sine and cosine, and that emerges because of a key identity. For all x, kosh squared minus cinch squared is equal to 1. And we can prove this by using the definitions of the hyperbolic sine and cosine and a little bit of algebra. And it's something you should be able to prove. Since we're in calculus, one of the questions we want to ask is, what's the derivative? So if we want to find the derivative of cinch, we'll use our definitions. Cinch is e to the x minus e to the minus x over 2, and the derivative is, and at this point we want to ask ourselves, self, have we seen this before? And in fact, yes, yes we have. This is kosh x. And we can find the derivative of kosh x in the same way, and this gives us our derivatives of the hyperbolic functions. We can define the inverse hyperbolic functions like we defined the inverse trigonometric functions. Inverse cinch is not a problem. y equals inverse cinch if and only if cinch y equals x. And this isn't a problem because there are no branches to worry about. Inverse kosh does have branches to worry about, and so we'll define y equals inverse kosh of x, if and only if kosh of y equals x, and y is greater than or equal to zero. What about the derivative of the inverse hyperbolic sine? So by definition, if y equals the inverse hyperbolic sine, then x is the hyperbolic sine of y. And this allows us to use implicit differentiation because our original function was given in terms of x, our derivative should also be in terms of x, so we can use our Pythagorean identity and solve for kosh y, where we take only the positive square root because kosh must be a positive number, and that gives us our derivative of inverse hyperbolic sine.