Derivative of sin(z^6)+sin^6(z)

1. Differentiate $\sin^{6}{\left(z \right)} + \sin{\left(z^{6} \right)}$ term by term:

   1. Let $u = z^{6}$.

   2. The derivative of sine is cosine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d z} z^{6}$:

      1. Apply the power rule: $z^{6}$ goes to $6 z^{5}$

      The result of the chain rule is:

      $$6 z^{5} \cos{\left(z^{6} \right)}$$

   4. Let $u = \sin{\left(z \right)}$.

   5. Apply the power rule: $u^{6}$ goes to $6 u^{5}$

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

      1. The derivative of sine is cosine:

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

      The result of the chain rule is:

      $$6 \sin^{5}{\left(z \right)} \cos{\left(z \right)}$$

   The result is: $6 z^{5} \cos{\left(z^{6} \right)} + 6 \sin^{5}{\left(z \right)} \cos{\left(z \right)}$

The answer is:

$$6 z^{5} \cos{\left(z^{6} \right)} + 6 \sin^{5}{\left(z \right)} \cos{\left(z \right)}$$