$$$\frac{z_{0}^{4}}{1 - z_{0}}$$$の導関数

$$$f{\left(z_{0} \right)} = z_{0}^{4}$$$ と $$$g{\left(z_{0} \right)} = 1 - z_{0}$$$ に対して商の微分法則 $$$\frac{d}{dz_{0}} \left(\frac{f{\left(z_{0} \right)}}{g{\left(z_{0} \right)}}\right) = \frac{\frac{d}{dz_{0}} \left(f{\left(z_{0} \right)}\right) g{\left(z_{0} \right)} - f{\left(z_{0} \right)} \frac{d}{dz_{0}} \left(g{\left(z_{0} \right)}\right)}{g^{2}{\left(z_{0} \right)}}$$$ を適用する：

$${\color{red}\left(\frac{d}{dz_{0}} \left(\frac{z_{0}^{4}}{1 - z_{0}}\right)\right)} = {\color{red}\left(\frac{\frac{d}{dz_{0}} \left(z_{0}^{4}\right) \left(1 - z_{0}\right) - z_{0}^{4} \frac{d}{dz_{0}} \left(1 - z_{0}\right)}{\left(1 - z_{0}\right)^{2}}\right)}$$

和/差の導関数は、導関数の和/差である:

$$\frac{- z_{0}^{4} {\color{red}\left(\frac{d}{dz_{0}} \left(1 - z_{0}\right)\right)} + \left(1 - z_{0}\right) \frac{d}{dz_{0}} \left(z_{0}^{4}\right)}{\left(1 - z_{0}\right)^{2}} = \frac{- z_{0}^{4} {\color{red}\left(\frac{d}{dz_{0}} \left(1\right) - \frac{d}{dz_{0}} \left(z_{0}\right)\right)} + \left(1 - z_{0}\right) \frac{d}{dz_{0}} \left(z_{0}^{4}\right)}{\left(1 - z_{0}\right)^{2}}$$

定数の導数は$$$0$$$です:

$$\frac{- z_{0}^{4} \left({\color{red}\left(\frac{d}{dz_{0}} \left(1\right)\right)} - \frac{d}{dz_{0}} \left(z_{0}\right)\right) + \left(1 - z_{0}\right) \frac{d}{dz_{0}} \left(z_{0}^{4}\right)}{\left(1 - z_{0}\right)^{2}} = \frac{- z_{0}^{4} \left({\color{red}\left(0\right)} - \frac{d}{dz_{0}} \left(z_{0}\right)\right) + \left(1 - z_{0}\right) \frac{d}{dz_{0}} \left(z_{0}^{4}\right)}{\left(1 - z_{0}\right)^{2}}$$

$$$n = 1$$$ を用いて冪法則 $$$\frac{d}{dz_{0}} \left(z_{0}^{n}\right) = n z_{0}^{n - 1}$$$ を適用すると、すなわち $$$\frac{d}{dz_{0}} \left(z_{0}\right) = 1$$$:

$$\frac{z_{0}^{4} {\color{red}\left(\frac{d}{dz_{0}} \left(z_{0}\right)\right)} + \left(1 - z_{0}\right) \frac{d}{dz_{0}} \left(z_{0}^{4}\right)}{\left(1 - z_{0}\right)^{2}} = \frac{z_{0}^{4} {\color{red}\left(1\right)} + \left(1 - z_{0}\right) \frac{d}{dz_{0}} \left(z_{0}^{4}\right)}{\left(1 - z_{0}\right)^{2}}$$

冪法則 $$$\frac{d}{dz_{0}} \left(z_{0}^{n}\right) = n z_{0}^{n - 1}$$$ を $$$n = 4$$$ に対して適用する:

$$\frac{z_{0}^{4} + \left(1 - z_{0}\right) {\color{red}\left(\frac{d}{dz_{0}} \left(z_{0}^{4}\right)\right)}}{\left(1 - z_{0}\right)^{2}} = \frac{z_{0}^{4} + \left(1 - z_{0}\right) {\color{red}\left(4 z_{0}^{3}\right)}}{\left(1 - z_{0}\right)^{2}}$$

簡単化せよ:

$$\frac{z_{0}^{4} + 4 z_{0}^{3} \left(1 - z_{0}\right)}{\left(1 - z_{0}\right)^{2}} = \frac{z_{0}^{3} \left(4 - 3 z_{0}\right)}{\left(z_{0} - 1\right)^{2}}$$

したがって、$$$\frac{d}{dz_{0}} \left(\frac{z_{0}^{4}}{1 - z_{0}}\right) = \frac{z_{0}^{3} \left(4 - 3 z_{0}\right)}{\left(z_{0} - 1\right)^{2}}$$$。