Derivative of 5cos(4x^2)

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

   1. Let $u = 4 x^{2}$.

   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 $\frac{d}{d x} 4 x^{2}$:

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

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

         So, the result is: $8 x$

      The result of the chain rule is:

      $$- 8 x \sin{\left(4 x^{2} \right)}$$

   So, the result is: $- 40 x \sin{\left(4 x^{2} \right)}$

The answer is: