Derivative of z*sin(z)

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 $1$

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

   1. The derivative of sine is cosine:

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

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

The answer is:

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