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Determine the derivative.$$f(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$

$\frac{4}{\left(e^{x}+e^{-x}\right)^{2}}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

Campbell University

Harvey Mudd College

Idaho State University

Lectures

01:21

Determine the derivative.<…

01:35

Find the derivative of the…

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01:53

02:21

So we're task of finding the derivative of this giant quotient. So as we look at this problem where we have e to the x minus, eat the *** rex all over a to the x plus e to the negative x. So, as I mentioned, we have this quotient just telling us we're going to do the quotient rule. But while I'm at it, Amounts will also mention that we're going to have a chain bull for these negative eso. It's kind of jump right into it where the question will tells us to take the derivative of the top. Well, the derivative of E to the X is e to the X, but the derivative of E to the negative X is e to the negative X times negative one because you have to do the derivative of negative X, which is negative one. So I'm just gonna change the sign of that. Yeah, and the denominator stays the same. But if you notice the denominator is the same as what we just found the derivative of, so you can multiply by it again or you can write squared and then minus the derivative of the bottom, I would be e to the X on Ben. The directive of E to the negative X is e to the negative X times negative one. So instead of having a plus, they're changing to minus. But even that is multiplied by leaving the top alone. So it's squared all over the denominator square. Now, some teachers will let you leave your answer like this. Mhm. However, what would actually be advantageous? So I don't know. I'll circle it because some teachers will let you leave your answer like that. Otherwise, what you should do is actually, I guess, foil that out. You know where either the X times e t s, you add your exponents would be eat of the two x on. Then there'd be plus two e to the X E to the negative x mhm thinking if you could actually figure that out a little bit faster on then, minus the quantity of e 22 works, Uh, there'll be another minus a negative to either the exceeded Negative X and then a native times. The name is positive, but again, you have to distribute this negative, so it becomes a minus. Eat native to X and the denominator stays the same, but I should point out a few things. The first thing would be that we have to eat of the two X and then later, minus into the two x e to the negative two x minus E to the negative two X Now this other piece right here remember? And I said it. If you multiply with the same base, you add the exponents well, X plus negative X is zero or zero X and anything to the zero powers. One. So, basically what it boils down to is that's two plus two in the numerator. So if you're going to simplify, go ahead and just simplify. The 42 plus two is four, uh, over eight of the X plus e to the negative X squared. Hopefully, all of that made sense.

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