Derivative of (-ln2)cos((log2x))

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

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

   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 $\frac{d}{d x} \log{\left(2 x \right)}$:

      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 of the chain rule is:

      $$- \frac{\sin{\left(\log{\left(2 x \right)} \right)}}{x}$$

   So, the result is: $\frac{\log{\left(2 \right)} \sin{\left(\log{\left(2 x \right)} \right)}}{x}$

The answer is:

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