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Chapter 9: Problem 4

Find the differential \(d y\). \(y=\frac{x+1}{2 x-1}\)

Short Answer

Expert verified
The differential \(d y\) of the function \(y=\frac{x+1}{2 x-1}\) is \(-\frac{3}{(2x-1)^2}\.d x\).

Step by step solution

01

Identify \(u\) and \(v\)

In our function \(y=\frac{x+1}{2 x-1}\), \(u\) can be identified as \(x+1\) and \(v\) as \(2x-1\).
02

Compute the Derivatives \(u'\) and \(v'\)

Compute the derivative of \(u\) with respect to \(x\) to get \(u' = 1\). Next, compute the derivative of \(v\) with respect to \(x\) to get \(v' = 2\).
03

Apply the Quotient Rule

Now that we have \(u\), \(v\), \(u'\), and \(v'\), we can apply the quotient rule \(\frac{vu' - uv'}{v^2}\) which grants us \(\frac{(2x-1)(1)-(x+1)(2)}{(2x-1)^2}\). This simplifies to \(\frac{-3}{(2x-1)^2}\).
04

Compute the Differential \(d y\)

To find the differential \(d y\), we just have to multiply the derivative by \(d x\). So, \(d y = -\frac{3}{(2x-1)^2}\.d x\).

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Most popular questions from this chapter

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The management of a company is considering three possible models for predicting the company's profits from 2003 through 2008 . Model I gives the expected annual profits if the current trends continue. Models II and III give the expected annual profits for various combinations of increased labor and energy costs. In each model, \(p\) is the profit (in billions of dollars) and \(t=0\) corresponds to 2003 . Model I: \(\quad p=0.03 t^{2}-0.01 t+3.39\) Model II: \(\quad p=0.08 t+3.36\) Model III: \(p=-0.07 t^{2}+0.05 t+3.38\) (a) Use a graphing utility to graph all three models in the same viewing window. (b) For which models are profits increasing during the interval from 2003 through 2008 ? (c) Which model is the most optimistic? Which is the most pessimistic? Which model would you choose? Explain.
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