Question

Find the derivatives of the functions:$f\left(x\right)=x^2{\log }_2\left(x{2}^x\right)$

Short answer

The required answer is$3x^2+2x{\log }_{2}x+\frac{x}{\ln 2}$

Step-by-step solution

Step 1. Given information. 

The given function is:$f\left(x\right)=x^2{\log }_2\left(x{2}^x\right)$

Step 2. Find the derivative of the given function.

$$\begin{gathered} f\left(x\right)=x^2{\log }_2\left(x{2}^x\right) \\ f\text{'}\left(x\right)=\frac{d}{dx}\left[x^2{\log }_2\left(x{2}^x\right)\right] \\ =\frac{d}{dx}{x}^2\left[{\log }_{2}x+{\log }_2\left(2^x\right)\right] \\ =\frac{d}{dx}{x}^2\left[{\log }_{2}x+x\right] \\ =\frac{d}{dx}{x}^3+\frac{d}{dx}{x}^2{\log }_{2}x \\ =3x^2+2x{\log }_{2}x+x^2\frac{1}{x\ln 2} \\ =3x^2+2x{\log }_{2}x+\frac{x}{\ln 2} \end{gathered}$$