Question
Given \( y=\sqrt[3]{1-5 z} \), find \( \frac{d y}{d z} \) \( \frac{d y}{d z}= \)
Ask by Frazier Murphy. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\frac{dy}{dz} = -\frac{5}{3} (1 - 5z)^{-\frac{2}{3}}
\]
Solution
To find the derivative \( \frac{dy}{dz} \) of the function \( y = \sqrt[3]{1 - 5z} \), follow these steps:
1. **Express \( y \) in exponential form:**
\[
y = (1 - 5z)^{\frac{1}{3}}
\]
2. **Differentiate using the chain rule:**
\[
\frac{dy}{dz} = \frac{1}{3} (1 - 5z)^{-\frac{2}{3}} \cdot (-5)
\]
3. **Simplify the expression:**
\[
\frac{dy}{dz} = -\frac{5}{3} (1 - 5z)^{-\frac{2}{3}}
\]
**Final Answer:**
\[
\frac{d y}{d z} = -\frac{5}{3}\, (1 - 5 z)^{-\, \frac{2}{3}}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To differentiate \( y = \sqrt[3]{1 - 5z} \), we can rewrite it as \( y = (1 - 5z)^{1/3} \). Now applying the chain rule: \[ \frac{d y}{d z} = \frac{1}{3}(1 - 5z)^{-2/3} \cdot (-5) \] Simplifying this gives: \[ \frac{d y}{d z} = -\frac{5}{3(1 - 5z)^{2/3}} \] So, the final answer is: \[ \frac{d y}{d z} = -\frac{5}{3(1 - 5z)^{2/3}} \]
