 So with the six hyperbolic functions now in hand, we can talk about their derivatives How would we compute the derivatives of these hyperbolic functions? Well, it turns out the derivatives We're gonna see are very similar to their Trignometric counterparts for example consider the derivative of cinch We're gonna see in just a second the derivative of cinch is equal to kosh This is actually a fairly simple argument here if you want to take the derivative of cinch Well, then basically We just have to take the derivative of its applied definition We get e to the x minus e to the negative x over 2 prime We can factor out the one half so we get e to the x minus e to the negative x prime Taking the derivatives of these things separately the derivative of e to the x is itself and the derivative of e to the negative x Well by the chain rule that's just gonna equal negative e to negative x So we get one half e to the x minus a negative e to the x negative x excuse me But that double negative becomes a positive so we end up with e to the x plus e to the negative x over 2 That's just the same thing as kosh So we see that the derivative of cinch is equal to kosh The sim a similar statements can also be true for kosh, right? We're gonna what we could prove by the same argument that the derivative of kosh is equal to cinch Now you'll notice that the negative sign is missing. This is something we saw earlier when we talked about hyperbolic functions It's basically the same stuff as trigonometric functions, but sometimes the negatives become positive than positive become negatives So the derivative of cinch is positive kosh and the derivative of kosh is positive cinch You could prove it by a similar argument as we just did with cinch How about the derivative of of hyperbolic tangent the derivative hyperbolic tangent is gonna be hyperbolic secant squared It's basically the same argument if you want to take the derivative of hyperbolic Tangent well, you treat it like it's a quotient. You're gonna take cinch Over kosh and you take the derivative via the quotient rule. So you're gonna get low D high the derivative of cinch is kosh Minus high D low the derivative of kosh is a cinch Squared the bottom here we go. So we get this kosh squared on the bottom You'll notice of course in the numerator that we have a double kosh. That's a kosh squared We get a double cinch. That's a cinch squared. So we get kosh squared minus cinch squared Now by the Pythagorean identity on Hyperbolic functions kosh squared minus cinch squared is equal to one So this thing will just become a one over kosh squared, which is the same thing as hyperbolic secant squared And so the derivative of hyperbolic tangents this is basically the exact same thing It's just we have to add in these h's into consideration when you take the derivative of hyperbolic Cotangent, it's basically the same thing. You're gonna get a negative hyperbolic kosh secant squared When you take the derivative of hyperbolic secant, you're gonna get a negative Hyperbolic secant times tangent there. So there's this extra negative sign so you have to watch out for that When you take the derivative of hyperbolic kosh secant, you're gonna get a negative hyperbolic kosh secant hyperbolic hyperbolic cotangent. So Basically we just add a bunch of h's everywhere. The only difference is going to be here and here Where this here is a consequence of this one right here So if you can remember that the derivative of kosh is cinch You're basically going to be just fine when you calculate the derivatives of these things. The calculating derivatives of hyperbolic functions Is very similar to trigonometric functions. Just be cautious of that negative sign So basically we have a brand new Evolution a brand new generation of Pokemon that can be entered into our decks into our team now So welcome Pokemon cinch and kosh edition Because we can take all of the functions. We already know trig functions, logarithms, exponentials, power functions with all the rules We know power rule, Quotient rule, product rule, chain rule. We can combine those all together with the hyperbolic functions and calculate their derivatives So could we calculate the derivative of kosh of the square root of x? Absolutely. We should recognize that there's two functions in play. There's the square root of x sitting inside of kosh And so we take the derivative by the chain rule the derivative of kosh of the square root of x take the outer derivative You're gonna get the derivative of kosh, which we saw earlier with cinch cinch of the square root of x Then you're gonna multiply that by the inner derivative which is the derivative of the square root of x Which is one over two times the square root of x and so that's basically it not much more You can do with it really. You can write this as cinch square root of x over Two times the square root of x again, no simplifications going on here the derivative calculations really no more different than calculating derivatives of the trigonometric functions again Just remember derivative of cinch is kosh derivative of kosh of cinch and you're gonna be just fine