Derivative of 2cos^4(x)-4x^2

1. Differentiate $- 4 x^{2} + 2 \cos^{4}{\left(x \right)}$ term by term:

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

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

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

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

         1. The derivative of cosine is negative sine:

            $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

         The result of the chain rule is:

         $$- 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$$

      So, the result is: $- 8 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$

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

      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: $8 x$

      So, the result is: $- 8 x$

   The result is: $- 8 x - 8 \sin{\left(x \right)} \cos^{3}{\left(x \right)}$

The answer is: