Q 35. - Calculus

Question

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

Step-by-Step Solution

Answer

The required answer is $3x^2+2x{\log }_2 x+\frac{x}{\ln 2}$

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.

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