 Now as a general rule, we can define new functions in terms of old functions. For example, f of x equals e raised to the power of sine 3x. Generally it's not worth giving the functions a new name, but sometimes it's convenient. And so we define the hyperbolic sine and hyperbolic cosine function as follows, hyperbolic sine e to the x minus e to the minus x over 2 and hyperbolic cosine e to the x plus e to the minus x over 2. And one more common thing we do, we should say hyperbolic sine and hyperbolic cosine, but we don't. We do write them this way with an h at the end, but h's at the end are typically silent, so we generally pronounce hyperbolic sine as sinh and hyperbolic cosine as cosh. Now if you study mathematics for any length of time, you realize something very important. Mathematicians are terrible at coming up with new names. In fact, these new functions are called the hyperbolic sine and cosine. So you might wonder, why hyperbolic sine and cosine? And here's another good rule. If a mathematician names something for something else, there's probably some sort of connection or at least a similarity between the two objects. And in this case, that connection emerges because of an important identity. For all x, the square of the hyperbolic cosine minus the square of the hyperbolic sine 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 sinh, we'll use our definitions. Sinh 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 cosh x. And we can find the derivative of cosh x in the same way, and this gives us our derivatives of the hyperbolic functions. We can define the inverse hyperbolic functions like we define the inverse trigonometric functions. Inverse sinh isn't a problem. y equals inverse sinh if and only if sinh y equals x. And this isn't a problem because there are no branches to consider. Inverse cosh does have branches to worry about, and so we'll define y equals inverse cosh of x only if cosh of y equals x and y is greater than or equal to zero. How about the derivative? So by definition, if y equals the inverse hyperbolic sinh, then x is the hyperbolic sinh 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 cosh y, where we take only the positive square root because cosh must be a positive number, and that gives us our derivative of inverse hyperbolic sinh.