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Compute the indicated derivative using the chain rule. dy dx dy dx x = 1 + 3t, y = -7t; Need Help? Read It

Compute the indicated derivative using the chain rule. dy dx dy dx x = 1 + 3t, y = -7t; Need Help? Read It

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
**Compute the Indicated Derivative Using the Chain Rule** Given: - \( x = 1 + 3t \) - \( y = -7t \) Find: - \(\frac{dy}{dx}\) **Solution:** You are prompted to compute \(\frac{dy}{dx}\) using the relationships provided for \(x\) and \(y\) as functions of \(t\). **Steps:** 1. Compute \(\frac{dx}{dt}\). - \(\frac{dx}{dt} = \frac{d}{dt}(1 + 3t) = 3\) 2. Compute \(\frac{dy}{dt}\). - \(\frac{dy}{dt} = \frac{d}{dt}(-7t) = -7\) 3. Apply the chain rule to find \(\frac{dy}{dx}\): - \(\frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx}\) - Since \(\frac{dt}{dx} = \frac{1}{\frac{dx}{dt}}\), we have \(\frac{dt}{dx} = \frac{1}{3}\) 4. Therefore, \(\frac{dy}{dx} = -7 \times \frac{1}{3} = -\frac{7}{3}\) **Conclusion:** - The derivative \(\frac{dy}{dx}\) is \(-\frac{7}{3}\). **Additional Resources:** If you need further assistance, there's a "Need Help?" section with a "Read It" button for more detailed guidance.
Transcribed Image Text:**Compute the Indicated Derivative Using the Chain Rule** Given: - \( x = 1 + 3t \) - \( y = -7t \) Find: - \(\frac{dy}{dx}\) **Solution:** You are prompted to compute \(\frac{dy}{dx}\) using the relationships provided for \(x\) and \(y\) as functions of \(t\). **Steps:** 1. Compute \(\frac{dx}{dt}\). - \(\frac{dx}{dt} = \frac{d}{dt}(1 + 3t) = 3\) 2. Compute \(\frac{dy}{dt}\). - \(\frac{dy}{dt} = \frac{d}{dt}(-7t) = -7\) 3. Apply the chain rule to find \(\frac{dy}{dx}\): - \(\frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx}\) - Since \(\frac{dt}{dx} = \frac{1}{\frac{dx}{dt}}\), we have \(\frac{dt}{dx} = \frac{1}{3}\) 4. Therefore, \(\frac{dy}{dx} = -7 \times \frac{1}{3} = -\frac{7}{3}\) **Conclusion:** - The derivative \(\frac{dy}{dx}\) is \(-\frac{7}{3}\). **Additional Resources:** If you need further assistance, there's a "Need Help?" section with a "Read It" button for more detailed guidance.
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