Derivative of y=sin^3x+cos^3x

1. Differentiate $\sin^{3}{\left(x \right)} + \cos^{3}{\left(x \right)}$ term by term:

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

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

   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:

      $$3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$$

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

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

   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:

      $$- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

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

2. Now simplify:

   $$- 3 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}$$

The answer is:

$$- 3 \sqrt{2} \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)}$$