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Chapter 12: TF. 1 (page 946)

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

Short Answer

Expert verified

$\frac{\partial f}{\partial x} = 2 x y^{3} \frac{\partial f}{\partial y} = 3 x^{2} y^{2}$

Step by step solution

01

Step 1. Given information

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

02

Step 2. Finding partial derivatives

$f (x, y) = x^{2} y^{3} \Rightarrow \frac{\partial f}{\partial x} = 2 x y^{3} \Rightarrow \frac{\partial f}{\partial y} = 3 x^{2} y^{2}$

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Most popular questions from this chapter

Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.

$\frac{\partial z}{\partial r} w h e n z = (x^{2} + x y) e^{y}, x = r \cos θ a n d y = r \sin θ$

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

$f (x, y) = x \sin y, P = (3, \frac{π}{2})$

In Exercises 24–32, find the maximum and minimum of the functionf subject to the given constraint. In each case explain why the maximum and minimum must both exist.

$f (x, y, z) = x y z when x^{2} + 4 y^{2} + 16 z^{2} = 64$

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.

$f (x, y) = x + y when x^{2} + 4 y^{2} = 16$

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.

$f (x, y, z) = x y z when x^{2} + y^{2} + z^{2} = 9$

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