Derivative of y=log(10)cos^24×

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

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

   2. Apply the power rule: $u^{24}$ goes to $24 u^{23}$

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

      1. The derivative of cosine is negative sine:

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

      The result of the chain rule is:

      $$- 24 \sin{\left(x \right)} \cos^{23}{\left(x \right)}$$

   So, the result is: $- 24 \log{\left(10 \right)} \sin{\left(x \right)} \cos^{23}{\left(x \right)}$

The answer is:

$$- 24 \log{\left(10 \right)} \sin{\left(x \right)} \cos^{23}{\left(x \right)}$$