TF. 5 A聽chain聽rule聽for聽functions聽... [FREE SOLUTION]

Chapter 12: TF. 5 (page 946)

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Find \frac{鈭� f}{鈭� s} and \frac{鈭� f}{鈭� t} using the preceding results .$

Short Answer

Expert verified

$\frac{鈭� f}{鈭� s} = 5 s^{4} \cos^{2} t \sin^{3} t \frac{鈭� f}{鈭� t} = - 2 s^{5} \cos t \sin t + 3 s^{5} \cos^{3} t \sin^{2} t$

Step by step solution

Step 1. Given information

$f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t .$

Step 2. Finding partial derivatives

$f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . 鈬� f (x, y) = s^{5} \cos^{2} t \sin^{3} t 1 . \frac{鈭� f}{鈭� s} = 5 s^{4} \cos^{2} t \sin^{3} t 2 . \frac{鈭� f}{鈭� t} = s^{5} (2 \cos t) (- \sin t) + s^{5} \cos^{2} t (3 \sin^{2} t) (\cos t) 鈬� \frac{鈭� f}{鈭� t} = - 2 s^{5} \cos t \sin t + 3 s^{5} \cos^{3} t \sin^{2} t$

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91影视

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Find \frac{鈭� f}{鈭� s} and \frac{鈭� f}{鈭� t} using the preceding results .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding partial derivatives

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91影视

Achainruleforfunctionsoftwovariables:Letf(x,y)=x2y3,x=scost,andy=ssint.Find鈭�f鈭�sand鈭�f鈭�tusingtheprecedingresults.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding partial derivatives

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Mechanics Maths

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Find \frac{鈭� f}{鈭� s} and \frac{鈭� f}{鈭� t} using the preceding results .$