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

1. Let $u = x \sin{\left(x \right)} + \cos{\left(x \right)}$.

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

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

   1. Differentiate $x \sin{\left(x \right)} + \cos{\left(x \right)}$ 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 $1$

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

         1. The derivative of sine is cosine:

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

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

      2. The derivative of cosine is negative sine:

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

      The result is: $x \cos{\left(x \right)}$

   The result of the chain rule is:

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

The answer is: