Derivative of сos2(x-7)

1. Let $u = \cos{\left(x - 7 \right)}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x - 7 \right)}$:

   1. Let $u = x - 7$.

   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 $\frac{d}{d x} \left(x - 7\right)$:

      1. Differentiate $x - 7$ term by term:

         1. Apply the power rule: $x$ goes to $1$

         2. The derivative of the constant $-7$ is zero.

         The result is: $1$

      The result of the chain rule is:

      $$- \sin{\left(x - 7 \right)}$$

   The result of the chain rule is:

   $$- 2 \sin{\left(x - 7 \right)} \cos{\left(x - 7 \right)}$$

4. Now simplify:

   $$- \sin{\left(2 x - 14 \right)}$$

The answer is:

$$- \sin{\left(2 x - 14 \right)}$$