Derivative of x*log(2*x)+2^sin(x)

1. Differentiate $2^{\sin{\left(x \right)}} + x \log{\left(2 x \right)}$ term by term:

   1. Apply the product rule:

      $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

      $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Apply the power rule: $x$ goes to $1$

      $g{\left(x \right)} = \log{\left(2 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

      1. Let $u = 2 x$.

      2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

         1. The derivative of a constant times a function is the constant times the derivative of the function.

            1. Apply the power rule: $x$ goes to $1$

            So, the result is: $2$

         The result of the chain rule is:

         $$\frac{1}{x}$$

      The result is: $\log{\left(2 x \right)} + 1$

   2. Let $u = \sin{\left(x \right)}$.

   3. $\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$

   4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$:

      1. The derivative of sine is cosine:

         $$\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$$

      The result of the chain rule is:

      $$2^{\sin{\left(x \right)}} \log{\left(2 \right)} \cos{\left(x \right)}$$

   The result is: $2^{\sin{\left(x \right)}} \log{\left(2 \right)} \cos{\left(x \right)} + \log{\left(2 x \right)} + 1$

2. Now simplify:

   $$\log{\left(2^{2^{\sin{\left(x \right)}} \cos{\left(x \right)}} \right)} + \log{\left(2 x \right)} + 1$$

The answer is:

$$\log{\left(2^{2^{\sin{\left(x \right)}} \cos{\left(x \right)}} \right)} + \log{\left(2 x \right)} + 1$$