Derivative of sin(lnx)+c

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

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

   2. The derivative of sine is cosine:

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

   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:

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

   4. The derivative of the constant $c$ is zero.

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

The answer is:

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