Question

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

$f\left(x\right)=12x\sqrt[3]{1-x}\sqrt{x+1}$

Short answer

The derivative of function is$y\text{'}=12x\sqrt[3]{1-x}\sqrt{x+1}\left[\frac{1}{x}-\frac{1}{3\left(1-x\right)}+\frac{1}{2(x+1}\right]$

Step-by-step solution

Step 1. Given Information 

The given function is$f\left(x\right)=12x\sqrt[3]{1-x}\sqrt{x+1}$

Step 2. Calculation   

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

$$\begin{gathered} \ln y=\ln 12x\sqrt[3]{1-x}\sqrt{x+1} \\ \ln y=\ln \left(12x\right)+\ln \left[\sqrt[3]{1-x}\right]+\ln \sqrt{x+1} \\ \ln y=\ln \left(12x\right)+\ln {\left[1-x\right]}^{\frac{1}{3}}+\ln {(x+1)}^{\frac{1}{2}} \\ \ln y=\ln \left(12\right)+\ln \left(x\right)+\frac{1}{3}\ln (1-x)+\frac{1}{2}\ln (x+1) \\ \frac{1}{y}y\text{'}=0+\frac{1}{x}-\frac{1}{3(1-x)}+\frac{1}{2(x+1)} \\ y\text{'}=12x\sqrt[3]{1-x}\sqrt{x+1}\left[\frac{1}{x}-\frac{1}{3(1-x)}+\frac{1}{2(x+1)}\right] \end{gathered}$$