Find the derivative of the function. \[ f(z)=e^{z /(z-8)} \] \( f^{\prime}(z)= \)

To find the derivative of the function \( f(z)=e^{z /(z-8)} \), we can utilize the chain rule. First, we identify the outer function as \( e^u \) where \( u = \frac{z}{z-8} \). The derivative of \( e^u \) is \( e^u \cdot u' \). Now, we need to calculate \( u' \): \( u = \frac{z}{z-8} \) Using the quotient rule: \[ u' = \frac{(z-8) \cdot 1 - z \cdot 1}{(z-8)^2} = \frac{z - 8 - z}{(z-8)^2} = \frac{-8}{(z-8)^2} \] Now substituting into the derivative formula, we get: \[ f'(z) = e^{z/(z-8)} \cdot u' = e^{z/(z-8)} \cdot \frac{-8}{(z-8)^2} \] Thus, the derivative is: \[ f^{\prime}(z) = -\frac{8 e^{z/(z-8)}}{(z-8)^2} \]