Derivative of cos(10x)

1. Let $u = 10 x$.

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 $\frac{d}{d x} 10 x$:

   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: $10$

   The result of the chain rule is:

   $$- 10 \sin{\left(10 x \right)}$$

The answer is:

$$- 10 \sin{\left(10 x \right)}$$