Derivative of cos((a*x)+b)

1. Let $u = a x + b$.

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 $\frac{\partial}{\partial x} \left(a x + b\right)$:

   1. Differentiate $a x + b$ 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: $a$

      2. The derivative of the constant $b$ is zero.

      The result is: $a$

   The result of the chain rule is:

   $$- a \sin{\left(a x + b \right)}$$

4. Now simplify:

   $$- a \sin{\left(a x + b \right)}$$

The answer is:

$$- a \sin{\left(a x + b \right)}$$