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Problem 35 Hard Difficulty

Patients undergo dialysis treatment to remove urea front their blood when their kidneys are not functioning properly. Blood is diverted from the patient through a machine that filters out urea. Under certain conditions, the duration of dialysis required given that the initial urea concentration is $ c > 1, $ is given by the equation
$ t = \ln (\frac {3c + \sqrt {9c^2 - 8c}}{2} $
calculate the derivative of $ t $ with respect to $ c $ and interpret it.

Answer

$\frac{3 \sqrt{9 c^{2}-8 c}+9 c-4}{\sqrt{9 c^{2}-8 c}(3 c+\sqrt{9 c^{2}-8 c})}$

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Video Transcript

the equation were given Here in this problem is the time that a patient spins on dialysis as a function of the concentration of Yuria in the blood, and we're going to find its derivative. And the derivative is going to be how the amount of time spent in dialysis changes as the concentration of Yuria changes. So that's the interpretation. So let's go ahead and do the calculation. But first of all, if you think about the log properties that you've learned in the past, you can use the quotient property of logarithms to change how this equation looks before you differentiate, and that makes it a little bit easier. So the log of a quotient is a log minus a log so we can change this to the natural log of the numerator, minus the natural log of the denominator. And because the natural log of two is just a constant when we differentiate, it's just going to be zero. So the derivative of tea, remember the derivative of natural log is one over the inside times, the derivative of the inside. So we would have won over three C plus the square root of nine C squared minus eight C multiplied by the derivative of that inside, and that would be three plus 1/2 times a quantity. Nine C squared, minus eight C to the negative, 1/2 times the derivative of the inside of that, which would be 18 C minus eight. Okay, we can clean that up a little bit. I don't think it's necessary to simplify it, a huge amount, but what we could Dio is right it as three times three plus and now notice that we have a 1/2 and we have another factor that has two even numbers in it. So let's multiply the 18 and the eight by 1/2. So we have three plus the quantity nine C minus four over the square root of nine C squared minus eight C. And that whole thing is over three C plus the square root of nine C squared, minus eight seat. And I know that we do sometimes simplify things like this. But for the sake of this problem, I don't think we necessarily have to simplify that again. That tells us how the amount of time and dialysis is changing with respect to the amount of Yuria and the blood