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Numerade Educator

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Problem 35 Hard Difficulty

Find the derivative of the function.
$ y = \cos (\frac {1 - e^{2x}}{1 + e^{2x}}) $

Answer

$=\frac{4 e^{2 x}}{\left(1+e^{2 x}\right)^{2}} \cdot \sin \left(\frac{1-e^{2 x}}{1+e^{2 x}}\right)$

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Video Transcript

here we have a composite function, a function inside of function, and so we're going to find the derivative using the chain rule. But on the inside we have a quotient. So we're going to end up using the quotient rule at some point as well. So, first of all, for the derivative of the outside, the derivative of co sign is negative sign. So have negative sign of the quotient that we see. And then we're gonna multiply that by the derivative of the quotient. So we have the bottom one plus e to the two x times the derivative of the top and the derivative of one minus C to the two X would be minus or negative each of the two x Times two. We had to use the chain rule on that as well. The derivative of two X is too. So that's the bottom times the derivative of the top minus the top one minus each of the two x times, the derivative of the bottom. And that would be each of the two x Times two. And that's gonna be over the bottom squared. So over one plus e to the two x squared now what we want to do is see what we can do to simplify that second numerator, and I suspect that what we might do is factor some things out. So let's see what we could factor out of both. So we have a negative that we could factor out of both. We have a two that we could factor out of both, and we have a knee to the two X that we could factor out of both. So when we do that, we're going to have because we're factoring out of negative. It's going to cancel with the negative on the outsides. We're gonna have positive to each of the two X times the sign of one minus each of the two X over one plus each of the two X times. Now let's see what we have left. In that quotient, we have one plus E to the two x plus one minus each of the two X over one plus each of the two x squared. Now look what happens to the numerator of that quotient. The each of the two x and the minus each of the two x cancel, leaving us with just one plus one, and we know that that is to so let's rewrite her answer did ourselves a little more space. So what you see here is what we just had a moment ago, and we're going to take this to and multiply it by two. The two that we have in the front. So we have four e to the two x times. Or we could just write that over the denominator that we have here over one plus e to the two x squared times. A sign of the quotient. One minus each of the two x over one plus each of the two x.