Derivative of f(x)=5cos(1-5x)

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

   1. Let $u = 1 - 5 x$.

   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 $\frac{d}{d x} \left(1 - 5 x\right)$:

      1. Differentiate $1 - 5 x$ term by term:

         1. The derivative of the constant $1$ is zero.

         2. 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: $-5$

         The result is: $-5$

      The result of the chain rule is:

      $$- 5 \sin{\left(5 x - 1 \right)}$$

   So, the result is: $- 25 \sin{\left(5 x - 1 \right)}$

The answer is:

$$- 25 \sin{\left(5 x - 1 \right)}$$