Derivative of (x/2)+sin(log(x))

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

   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: $\frac{1}{2}$

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

   3. The derivative of sine is cosine:

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

   4. 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: $\frac{1}{2} + \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$

2. Now simplify:

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

The answer is: