Derivative of (z-2)(z+i)

1. Apply the product rule:

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

   $f{\left(z \right)} = z - 2$; to find $\frac{d}{d z} f{\left(z \right)}$:

   1. Differentiate $z - 2$ term by term:

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

      2. The derivative of the constant $-2$ is zero.

      The result is: $1$

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

   1. Differentiate $z + i$ term by term:

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

      2. The derivative of the constant $i$ is zero.

      The result is: $1$

   The result is: $2 z - 2 + i$

The answer is:

$$2 z - 2 + i$$