Derivative of x*(cos(logx)+sin(logx))

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 $1$

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

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

      1. Let $u = \log{\left(x \right)}$.

      2. The derivative of cosine is negative sine:

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

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

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

         The result of the chain rule is:

         $$- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$$

      4. Let $u = \log{\left(x \right)}$.

      5. The derivative of sine is cosine:

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

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

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

         The result of the chain rule is:

         $$\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$$

      The result is: $- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$

   The result is: $x \left(- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}\right) + \sin{\left(\log{\left(x \right)} \right)} + \cos{\left(\log{\left(x \right)} \right)}$

2. Now simplify:

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

The answer is: