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Ⓒ Differenitale y = √3x-7 X

Ⓒ Differenitale y = √3x-7 X

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Calculus Problem: Differentiation **Problem: 1. Differentiate \( y = \sqrt{3x - 7} \)** #### Solution Steps: 1. **Identify the function to differentiate:** \( y = \sqrt{3x - 7} \) 2. **Rewrite the function in exponential form:** \( y = (3x - 7)^{1/2} \) 3. **Apply the chain rule for differentiation:** - The chain rule states that the derivative of \( y \) with respect to \( x \) is \( dy/dx = (dy/du) \cdot (du/dx) \), where \( u = 3x - 7 \). 4. **Differentiate the outer function (power rule):** - Let \( u = 3x - 7 \) - Then, \( y = u^{1/2} \) - By the power rule, \( dy/du = \frac{1}{2}u^{-1/2} \) 5. **Differentiate the inner function:** - \( du/dx = 3 \) 6. **Combine the derivatives using the chain rule:** - \( dy/dx = (dy/du) \cdot (du/dx) \) - \( dy/dx = \frac{1}{2}u^{-1/2} \cdot 3 \) - Substitute back \( u = 3x - 7 \) - \( dy/dx = \frac{3}{2}(3x - 7)^{-1/2} \) 7. **Simplify the expression:** - \( dy/dx = \frac{3}{2\sqrt{3x - 7}} \) Thus, the derivative of \( y = \sqrt{3x - 7} \) with respect to \( x \) is \( \frac{3}{2\sqrt{3x - 7}} \). ### Conclusion: The differentiation problem demonstrates the application of the chain rule in finding the derivative of a composite function. This rule is essential for handling more complex differentiation tasks in calculus. Always ensure to express the final solution in its simplest form.
Transcribed Image Text:### Calculus Problem: Differentiation **Problem: 1. Differentiate \( y = \sqrt{3x - 7} \)** #### Solution Steps: 1. **Identify the function to differentiate:** \( y = \sqrt{3x - 7} \) 2. **Rewrite the function in exponential form:** \( y = (3x - 7)^{1/2} \) 3. **Apply the chain rule for differentiation:** - The chain rule states that the derivative of \( y \) with respect to \( x \) is \( dy/dx = (dy/du) \cdot (du/dx) \), where \( u = 3x - 7 \). 4. **Differentiate the outer function (power rule):** - Let \( u = 3x - 7 \) - Then, \( y = u^{1/2} \) - By the power rule, \( dy/du = \frac{1}{2}u^{-1/2} \) 5. **Differentiate the inner function:** - \( du/dx = 3 \) 6. **Combine the derivatives using the chain rule:** - \( dy/dx = (dy/du) \cdot (du/dx) \) - \( dy/dx = \frac{1}{2}u^{-1/2} \cdot 3 \) - Substitute back \( u = 3x - 7 \) - \( dy/dx = \frac{3}{2}(3x - 7)^{-1/2} \) 7. **Simplify the expression:** - \( dy/dx = \frac{3}{2\sqrt{3x - 7}} \) Thus, the derivative of \( y = \sqrt{3x - 7} \) with respect to \( x \) is \( \frac{3}{2\sqrt{3x - 7}} \). ### Conclusion: The differentiation problem demonstrates the application of the chain rule in finding the derivative of a composite function. This rule is essential for handling more complex differentiation tasks in calculus. Always ensure to express the final solution in its simplest form.
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