Question
Find the derivative of the function. \[ f(z)=e^{z /(z-8)} \] \( f^{\prime}(z)= \)
Ask by Ruiz Mccoy. in the United States
Feb 24,2025
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Answer
The derivative of the function \( f(z) = e^{z/(z-8)} \) is:
\[ f'(z) = \frac{-8 e^{z/(z-8)}}{(z-8)^2} \]
Solution
To differentiate f(z) = e^(z/(z - 8)), we use the chain rule. Let u(z) = z/(z - 8). Then f(z) = e^(u(z)) and f '(z) = e^(u(z)) · u '(z).
Step 1: Differentiate u(z):
u(z) = z/(z - 8)
This is a quotient, so applying the quotient rule:
u '(z) = [(z - 8)·(d/dz of z) - z·(d/dz of (z - 8))] / (z - 8)²
= [(z - 8)·1 - z·1] / (z - 8)²
= (z - 8 - z) / (z - 8)²
= (-8) / (z - 8)²
Step 2: Apply the chain rule:
f '(z) = e^(z/(z - 8)) · [(-8)/(z - 8)²]
Thus, the derivative of the function is:
f '(z) = (-8 e^(z/(z - 8)))/(z - 8)².
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Simplify this solution Bonus Knowledge
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: \[ f^{\prime}(z) = -\frac{8 e^{z/(z-8)}}{(z-8)^2} \]
