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Problem 17 Easy Difficulty

The hyperbolic functions are defined as $\sinh x=\frac{1}{2}\left(e^{x}-e^{-x}\right)$ and $\cosh x=\frac{1}{2}\left(e^{x}+e^{-x}\right)$.
a. Prove $\frac{d(\sinh x)}{d x}=\cosh x$.
b. Prove $\frac{d(\cosh x)}{d x}=\sinh x$.
c. Prove $\frac{d(\tanh x)}{d x}=\frac{1}{(\cosh x)^{2}}$ if $\tanh x=\frac{\sinh x}{\cosh x}$.


a. $\frac{d}{d x}(\sinh x)=\frac{d}{d x}\left[\frac{1}{2}\left(e^{x}-e^{-x}\right)\right]$
=\frac{1}{2}\left(e^{x}+e^{-x}\right) \\
=\cosh x
b. $\frac{d}{d x}(\cosh x)=\frac{1}{2}\left(e^{t}-e^{-t}\right)$
$=\sinh x$
c. Since tanh $x=\frac{\sinh x}{\cosh x^{\prime}}$
$\begin{aligned} \frac{d}{d x}(\tanh x)=\frac{\left(\frac{d}{d x} \sinh x\right)(\cosh x)}{(\cosh x)^{2}} \\-\frac{(\sinh x)\left(\frac{d}{d x} \cosh x\right)}{(\cosh x)^{2}} \\=& \frac{\frac{1}{2}\left(e^{x}+e^{-x}\right)\left(\frac{1}{2}\right)\left(e^{x}+e^{-x}\right)}{(\cosh x)^{2}} \\=& \frac{\frac{1}{4}(4)}{(\cosh x)^{2}} \\=& \frac{1}{(\cosh x)^{2}} \end{aligned}$


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Video Transcript

{'transcript': "Kasan H squared X minus Sign H Squared IX equals one. And we want to confirm this identity. So first, over here we see that coastline hx were given the it equals e to the x plus east of the negative x all over, too, and were given that sine h X equals each of the X minus each of the negative picks all over too. Someone to start with my left hand side. So you, uh, coast on h squared X minus Sign H squared X, and we're gonna plug in our values. What are givens? So we have e to the X plus A to the negative x all over, too. All squared minus sign which is each of the ex honesty to the negative X all over too. All squared. So now we're gonna square the top and bottom. So have you two the explicit each of the negative x squared over seeing squared minus e to the x minus into the negative Becks squared all over two squared. So I just separated itself in bottom, squared the boat So now we'll have a common denominator. Ah two squared which equal sport. I want to square the top. So I'm gonna use is this formula here. We're going to the X plus y squared formula, and that's how I'm going to expand it to the already more room somewhere. Well, go down here all over for so we're gonna use top formula. So have I exit to X first to e to the X into the negative books, plus e to the negative to x with the first boil minus. I'm gonna do the 2nd 1 e two the two X There's gonna be minus this time, too. Into the ex into the negative. Becks. Close. You should be close plus e to the negative to x, and I'm going to expand the I want to just to beat the negative and the second part. So her you need to the two x plus two e to the X into the negative Becks. Plus, he's in the negative to x minus each of two. X plus two e to the X into the negative picks minus eats in the negative to ICs. We're in north. These for a deceased here. Prentiss seasonally and it's all over for So now we have each of the each of the two X and this native is it Sue? It's gonna subtract out. Same with this one. And this one in Maryland with two into the ex. He's a negative picks. So we know that using our exponents properties X to the A Times exit B equals X to the A plus B So there's an understood one here and understood one here and we're gonna add those exponents and one minus one equals zero. So this E part will become each of zero, which is one. So the council's out and that part counts is out. So now we're left with two plus two only counsel don't the e court, because that equals zero. So that counsel's out. And now we look what two plus two all over four and two plus two equals four over core and four divided by four equals one. So now we're gonna go to part B, So we have one minus tan H squared X equals ticket H squared X. So we're gonna rearrange it, and we're gonna at hand to boat. Sorry. So we get one equal seeking H squared X plus sand H Squared IX. Someone started the right hand side so we have seek it were in bloom Seek it H squared X plus 10 each square dicks. So we're going to find were given the It's an H equals son h X over co sign age picks So I'm gonna switch this apart. Sign h x times one over co sign Age X. They were given that sine h X is into the x minus e the negative picks over two and the one over co sign. We're just going to flip the fraction so the denominator becomes the numerator and a new Moretti becomes the denominator because we're just gonna flip the direction. So there's two in this to a council out and we're left with It's the X minus into the negative books over E to the X plus into the negative X. So we're going to use that for change it a little bit Gonna make the smaller over here. Hopefully you can see that later. We're gonna do seek it hx so we have seek it H picks, which is one over co sign a checks. So we're just gonna flip our coast or infraction so two over into the ex plus e to the negative IX ever gonna use that for speaking in a little? You're right, this Smalling when I was here. So now we started with the right hand side. So I'm going to plug in value. So I'm gonna plug in to over each of the Ex plus e to the negative Bix squared, plus the tangent, which will be a to the x minus into the negative x all over into the ex close aides in the negative. Becks. False. Weird. So what is great is sucking bottom. So it's two squared over You too. The ex plus into the negative x Not all these squared close into the x minus e It's in the negative. Necks all squared over. You need to the xt plus e to the negative fix all squared. So explain this for her four and we'll have a common denominator already. So using the same formula from before we know that xt plus why squared equals X squared plus two x y plus y squared. So that's why I want to expand the denominator when we got down here. Still her you to the two x plus two e to the x into the negative x plus any to the negative to X No numerator is gonna be the same thing. Soap. Its attraction so X minus y squared equals X squared minus two. Ex wife close wars earlier So it's essentially the same thing. But you have reminders to ex mines that positive Well, in a perfect stone here, so four plus e to the two x my last two e to the X seats of the negative x plus e to the negative to X. So now, as we said before, next to the A terms accidentally equals X to the A plus B So I'm gonna use it to cancel out this heart right here. This each of the works and it's it's a negative X that will become one one because one minus one is zero and each of the zero power Anything to zero powers. One Mr. Everything that happened down here to this part right here. Don't counsel out because it's one and I will continue to simple for expression. 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So you, uh, coach tangent h squared X minus co seek it. H squared X equals one. So we're gonna need co surgeon h X, And that equals one over tangent A chicks so you can refer back toe problem be. We know that change it. H x equals e to the X minus e It's the negative x over e to the X first e to the negative picks still Ko Sanja. It would just be the reciprocal of that. Remember, Not here. Quotation will just be the reciprocal of that. So her e to the x plus e to the magnet picks over easy X minus the needs of the negativity. We're also Nico seek it. So we have coast ekin h Squared IX, not a square just close to get h X and were given that that equals won over. Sign A chicks so we know that sign h x equals e to the X minus e to the negative x all over too. SoCo seek It will just be the reciprocal of that. So, huh two over each of the ex minus e to the negative picks we're gonna use wasn't useful are for solving. I want to make you smile, but you can still see so early, huh? 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No need to expand our numerator and went to the same thing. So expose y squared equals x cleared plus to explore, plus y squared. And that's what I'm gonna use to expand my numerator. So her e to the two works list to each of the ex feats of the negative picks plus e to the negative to ICs and minus two squared, which equal sport. So now we'll have We're gonna look here at this e to the X, and it's a negative X as we know before exiting a times exit B equals X to the A plus B so you can see the We haven't understood one in front of the ex here, So when one is one is zero and e to the zero equals one. So this little castle out and the same thing Same exact thing with that one. I'm just gonna experiment and Raider units of two x plus any to the negative to its minus four all over into the to Alex plus E to the negative to ics minus two. I was to have a plus to love here right here for that plus two. 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