Derivative of cos((5*x)^7)

1. Let $u = \left(5 x\right)^{7}$.

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 $\frac{d}{d x} \left(5 x\right)^{7}$:

   1. Let $u = 5 x$.

   2. Apply the power rule: $u^{7}$ goes to $7 u^{6}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

      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: $5$

      The result of the chain rule is:

      $$546875 x^{6}$$

   The result of the chain rule is:

   $$- 546875 x^{6} \sin{\left(\left(5 x\right)^{7} \right)}$$

4. Now simplify:

   $$- 546875 x^{6} \sin{\left(78125 x^{7} \right)}$$

The answer is:

$$- 546875 x^{6} \sin{\left(78125 x^{7} \right)}$$