Derivative of cos^4(7x)

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

2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

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

   1. Let $u = 7 x$.

   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 $\frac{d}{d x} 7 x$:

      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: $7$

      The result of the chain rule is:

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

   The result of the chain rule is:

   $$- 28 \sin{\left(7 x \right)} \cos^{3}{\left(7 x \right)}$$

The answer is:

$$- 28 \sin{\left(7 x \right)} \cos^{3}{\left(7 x \right)}$$