Derivative of sin^2x+cos^4x+5x

1. Differentiate $5 x + \left(\sin^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}\right)$ term by term:

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

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

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

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

         1. The derivative of sine is cosine:

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

         The result of the chain rule is:

         $$2 \sin{\left(x \right)} \cos{\left(x \right)}$$

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

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

      6. 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:

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

      The result is: $- 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)}$

   2. 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 is: $- 4 \sin{\left(x \right)} \cos^{3}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} + 5$

2. Now simplify:

   $$5 - \frac{\sin{\left(4 x \right)}}{2}$$

The answer is:

$$5 - \frac{\sin{\left(4 x \right)}}{2}$$