Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
**Problem #28:** Find \( y' \) if \( y = x^2 \sin^4 x + x \cos^{-2} x \). **Explanation:** - The problem involves finding the derivative of the function \( y \) with respect to \( x \). - The function \( y \) is composed of two parts: \( x^2 \sin^4 x \) and \( x \cos^{-2} x \). - The first term \( x^2 \sin^4 x \) includes a polynomial \( x^2 \) and the fourth power of the sine function \( \sin^4 x \). - The second term \( x \cos^{-2} x \) involves \( x \) divided by the square of the cosine function \( \cos^2 x \). **Steps for Differentiation:** 1. Apply the product rule to differentiate each term with respect to \( x \). 2. For the first term \( x^2 \sin^4 x \), use the product rule: \( u = x^2 \) and \( v = \sin^4 x \). 3. For the second term \( x \cos^{-2} x \), apply the product rule: \( u = x \) and \( v = \cos^{-2} x \). This involves using the chain rule to handle the trigonometric functions raised to a power. Feel free to reach out for further elaboration on solving this differentiation problem!
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Transcribed Image Text:**Problem #28:** Find \( y' \) if \( y = x^2 \sin^4 x + x \cos^{-2} x \). **Explanation:** - The problem involves finding the derivative of the function \( y \) with respect to \( x \). - The function \( y \) is composed of two parts: \( x^2 \sin^4 x \) and \( x \cos^{-2} x \). - The first term \( x^2 \sin^4 x \) includes a polynomial \( x^2 \) and the fourth power of the sine function \( \sin^4 x \). - The second term \( x \cos^{-2} x \) involves \( x \) divided by the square of the cosine function \( \cos^2 x \). **Steps for Differentiation:** 1. Apply the product rule to differentiate each term with respect to \( x \). 2. For the first term \( x^2 \sin^4 x \), use the product rule: \( u = x^2 \) and \( v = \sin^4 x \). 3. For the second term \( x \cos^{-2} x \), apply the product rule: \( u = x \) and \( v = \cos^{-2} x \). This involves using the chain rule to handle the trigonometric functions raised to a power. Feel free to reach out for further elaboration on solving this differentiation problem!
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