Derivative of -sin(10x)+cos(5x-1)

1. Differentiate $- \sin{\left(10 x \right)} + \cos{\left(5 x - 1 \right)}$ term by term:

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

      1. Let $u = 10 x$.

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

         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:

         $$10 \cos{\left(10 x \right)}$$

      So, the result is: $- 10 \cos{\left(10 x \right)}$

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

   3. The derivative of cosine is negative sine:

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

   4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x - 1\right)$:

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

         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: $5$

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

         The result is: $5$

      The result of the chain rule is:

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

   The result is: $- 5 \sin{\left(5 x - 1 \right)} - 10 \cos{\left(10 x \right)}$

2. Now simplify:

   $$- 5 \sin{\left(5 x - 1 \right)} - 10 \cos{\left(10 x \right)}$$

The answer is:

$$- 5 \sin{\left(5 x - 1 \right)} - 10 \cos{\left(10 x \right)}$$