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Determine the derivative.$$f(x)=\frac{e^{x}-1}{e^{x}+1}$$

$\frac{2 e^{x}}{\left(e^{x}+1\right)^{2}}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

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University of Michigan - Ann Arbor

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Determine the derivative.<…

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Find the derivative.$$…

our task is to find the derivative of this quotient. Question is the giant division. So we're going to do the question now. I would even so we're not even gonna do the chain rule because when you look at this problem, either the X minus one divided by either the X plus one. Um, there's no chain rule embedded in this problem. So let's just jump right into the question rule, which says to take the derivative of the top while the derivative of E to the X is e to the X, the derivative of negative on a zero. So nothing to write their leave the denominator alone, minus the derivative of the bottom, which is e to the X. The derivative of one is zero. So nothing too right there lead the top alone all over the denominator square. Now, some teachers will let you leave your answer like this, so I'll serve it. But this problem, you can actually distribute into the you know, both the numerator. Uh, yeah, well, just remember that when your distributing, you add your exponents when you're multiplying with same base, so x plus x will be two x plus e to the x minus. Uh, the X Times. Either they actually be another eat of two X. Yeah. Um, yeah, Multi. With same base. You add your extra explicit should be two X and then negative times Negative. Makes that. Plus, I didn't square that, did I, Um, all over the denominator squared still. But if you're going to simplify, then go ahead and simplify all the way where either the to actually cancel. If you look in the numerator, you have to e to the X over that term. Eso This is probably the best answer, but either one of these does work.

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