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Chapter 12: TF. 1 (page 946)

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

Short Answer

Expert verified

鈭�f鈭�x=2xy3鈭�f鈭�y=3x2y2

Step by step solution

01

Step 1. Given information

f(x,y)=x2y3,x=scost,andy=ssint.

02

Step 2. Finding partial derivatives

f(x,y)=x2y3鈬�鈭�f鈭�x=2xy3鈬�鈭�f鈭�y=3x2y2

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Most popular questions from this chapter

Explain how you could use the method of Lagrange multipliers to find the extrema of a function of two variables, f(x,y),subject to the constraint that (x,y)is on the boundary of the rectangle Rdefined bya鈮�x鈮�bandc鈮�y鈮�d.

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

f(x,y)=xywhenx2+y2=16

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.

dzdtwhenz=x3ey,x=sint,andy=cost

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:

f(x,y)=tan鈭�1yx

Describe the meanings of each of the following mathematical expressions:

鈭�f(x,y,z)
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