Derivative of z*sin(z^3)

1. Apply the product rule:

   $$\frac{d}{d z} f{\left(z \right)} g{\left(z \right)} = f{\left(z \right)} \frac{d}{d z} g{\left(z \right)} + g{\left(z \right)} \frac{d}{d z} f{\left(z \right)}$$

   $f{\left(z \right)} = z$; to find $\frac{d}{d z} f{\left(z \right)}$:

   1. Apply the power rule: $z$ goes to $1$

   $g{\left(z \right)} = \sin{\left(z^{3} \right)}$; to find $\frac{d}{d z} g{\left(z \right)}$:

   1. Let $u = z^{3}$.

   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 $\frac{d}{d z} z^{3}$:

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

      The result of the chain rule is:

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

   The result is: $3 z^{3} \cos{\left(z^{3} \right)} + \sin{\left(z^{3} \right)}$

The answer is:

$$3 z^{3} \cos{\left(z^{3} \right)} + \sin{\left(z^{3} \right)}$$