Question

Find the derivatives of the functions:

$f\left(x\right)=\ln \left(x^2{2}^x\right)$

Short answer

The required answer is$\frac{2}{x}+\ln 2$.

Step-by-step solution

Step 1. Given information. 

The given function is:$f\left(x\right)=\ln \left(x^2{2}^x\right)$

Step 2. Find the derivative of the given function.

$$\begin{gathered} f\left(x\right)=\ln \left(x^2{2}^x\right) \\ f\left(x\right)=\ln \left(x^2\right)+\ln \left(2^x\right) \\ f\left(x\right)=2\ln x+x\ln 2 \\ f\text{'}\left(x\right)=\frac{d}{dx}\left(2\ln x+x\ln 2\right) \\ =\frac{d}{dx}\left(2\ln x\right)+\frac{d}{dx}\left(x\ln 2\right) \\ =\frac{2}{x}+\ln 2 \end{gathered}$$