Derivative of cos(k*x^63)

1. Let $u = k x^{63}$.

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 $\frac{\partial}{\partial x} k x^{63}$:

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

      1. Apply the power rule: $x^{63}$ goes to $63 x^{62}$

      So, the result is: $63 k x^{62}$

   The result of the chain rule is:

   $$- 63 k x^{62} \sin{\left(k x^{63} \right)}$$

The answer is:

$$- 63 k x^{62} \sin{\left(k x^{63} \right)}$$