Chapter 12: TF. 5 (page 946)

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

Short Answer

Expert verified

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

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

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

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding partial derivatives

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

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Applied Mathematics

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Study anywhere. Anytime. Across all devices.

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