Question

Use logarithmic differentiation to find the derivatives of each of the functions in Exercises 49–58.

$f\left(x\right)=\sqrt{x\ln |2^x+1|}$

Short answer

The derivative of function is $y\text{'}=\frac{1}{2}\sqrt{x\ln |2^x+1|}\left[\frac{1}{x}+\frac{1}{\ln (2^x+1)}·\frac{1}{2^x+1}\ln 2·2^x\right]$

Step-by-step solution

Step 1. Given Information

The given function is$f\left(x\right)=\sqrt{x\ln |2^x+1|}$

Step 2. Calculation  

First, we will take the log on both sides and then differentiate it,

$$\begin{gathered} \ln y=\ln \sqrt{x\ln |2^x+1|} \\ \ln y=\frac{1}{2}\left[\ln \left(x\ln |2^x+1|\right)\right] \\ \ln y=\frac{1}{2}\left[\ln x+\mathrm{lnln}|2^x+1|\right] \\ \frac{1}{y}y\text{'}=\frac{1}{2}\left[\frac{1}{x}+\frac{1}{\ln (2^x+1)}\left(\frac{1}{2^x+1}\right)\ln 2\left(2^x\right)\right] \\ y\text{'}=\frac{1}{2}\sqrt{x\ln |2^x+1|}\left[\frac{1}{x}+\frac{1}{\ln (2^x+1)}\left(\frac{1}{2^x+1}\right)\ln 2\left(2^x\right)\right] \end{gathered}$$