Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
3.6-Number 50, please
# Differentiation Rules ## Problems and Exercises: ### 43. Given the function \( f(x) = cx + \ln (\cos \, x) \), find the value of \( c \) for which \( f'(\pi/4) = 6 \). **Answer**: ### 44. Let \( f(x) = \log_8 (3x^2 - 2) \). Determine the value of \( b \) for which \( f'(1) = 3 \). **Answer**: ### 45-56. Use logarithmic differentiation to find the derivative of the function. 45. \( y = (x^2 + 2)^{2} (x^4 + 4)^{4} \) **Answer**: 46. \( y = \frac{e^{-x} \cos^2 x}{x^2 + x + 1} \) 47. \( y = \sqrt{\frac{x - 1}{x^4 + 1}} \) **Answer**: 48. \( y = \sqrt{x e^{x^2 - x} (x + 1)^{2/3}} \) 49. \( y = x^x \) **Answer**: 50. \( y = x^{1/x} \) **Answer**: These exercises involve applying various differentiation techniques, particularly focusing on logarithmic differentiation, to find derivatives of complex functions.
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Transcribed Image Text:# Differentiation Rules ## Problems and Exercises: ### 43. Given the function \( f(x) = cx + \ln (\cos \, x) \), find the value of \( c \) for which \( f'(\pi/4) = 6 \). **Answer**: ### 44. Let \( f(x) = \log_8 (3x^2 - 2) \). Determine the value of \( b \) for which \( f'(1) = 3 \). **Answer**: ### 45-56. Use logarithmic differentiation to find the derivative of the function. 45. \( y = (x^2 + 2)^{2} (x^4 + 4)^{4} \) **Answer**: 46. \( y = \frac{e^{-x} \cos^2 x}{x^2 + x + 1} \) 47. \( y = \sqrt{\frac{x - 1}{x^4 + 1}} \) **Answer**: 48. \( y = \sqrt{x e^{x^2 - x} (x + 1)^{2/3}} \) 49. \( y = x^x \) **Answer**: 50. \( y = x^{1/x} \) **Answer**: These exercises involve applying various differentiation techniques, particularly focusing on logarithmic differentiation, to find derivatives of complex functions.
Expert Solution
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Step 1

To find : Derivative of the function y=x1x

Take logarithm to both sides, we get

logy=1xlogx …..(1)

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