/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Find the four second partial der... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 30

Find the four second partial derivatives of the following functions. $$f(x, y)=\cos x y$$

Short Answer

Expert verified
Question: Determine the four second partial derivatives of the function $$f(x, y) = \cos{xy}$$. Answer: The four second partial derivatives are: 1. $$f_{xx} = -y^2\cos{xy}$$ 2. $$f_{yy} = -x^2\cos{xy}$$ 3. $$f_{xy} = -x\cos{xy}$$ 4. $$f_{yx} = -y\cos{xy}$$

Step by step solution

01

Find partial derivative with respect to x (f_x)

Find the partial derivative of the function f(x, y) with respect to x: $$f_x = \frac{\partial f}{\partial x} = \frac{\partial \cos{xy}}{\partial x}$$ Using the chain rule: $$f_x = -\sin{xy} \cdot \frac{\partial (xy)}{\partial x} = -y\sin{xy}$$
02

Find partial derivative with respect to y (f_y)

Find the partial derivative of the function f(x, y) with respect to y: $$f_y = \frac{\partial f}{\partial y} = \frac{\partial \cos{xy}}{\partial y}$$ Using the chain rule: $$f_y = -\sin{xy} \cdot \frac{\partial (xy)}{\partial y} = -x\sin{xy}$$
03

Find second partial derivative with respect to x (f_xx)

Now find the second partial derivative of f(x, y) with respect to x: $$f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial f_x}{\partial x} = \frac{\partial(-y\sin{xy})}{\partial x}$$ Using the chain rule: $$f_{xx} = -y^2\cos{xy}$$
04

Find second partial derivative with respect to y (f_yy)

Next, find the second partial derivative of f(x, y) with respect to y: $$f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial f_y}{\partial y} = \frac{\partial(-x\sin{xy})}{\partial y}$$ Using the chain rule: $$f_{yy} = -x^2\cos{xy}$$
05

Find mixed partial derivative with respect to x and y (f_xy)

Now, find the mixed second partial derivative of f(x, y) with respect to x and y: $$f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial f_x}{\partial y} = \frac{\partial(-y\sin{xy})}{\partial y}$$ Using the chain rule: $$f_{xy} = -x\cos{xy}$$
06

Find mixed partial derivative with respect to y and x (f_yx)

Finally, find the mixed second partial derivative of f(x, y) with respect to y and x: $$f_{yx} = \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial f_y}{\partial x} = \frac{\partial(-x\sin{xy})}{\partial x}$$ Using the chain rule: $$f_{yx} = -y\cos{xy}$$ Thus, the four second partial derivatives are: 1. $$f_{xx} = -y^2\cos{xy}$$ 2. $$f_{yy} = -x^2\cos{xy}$$ 3. $$f_{xy} = -x\cos{xy}$$ 4. $$f_{yx} = -y\cos{xy}$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Maximizing a sum Find the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the condition that \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\).

Maximizing a sum Geometric and arithmetic means Prove that the geometric mean of a set of positive numbers \(\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n}\) is no greater than the arithmetic mean \(\left(x_{1}+\cdots+x_{n}\right) / n\) in the following cases. a. Find the maximum value of \(x y z,\) subject to \(x+y+z=k\) where \(k\) is a real number and \(x>0, y>0,\) and \(z>0 .\) Use the result to prove that $$(x y z)^{1 / 3} \leq \frac{x+y+z}{3}$$ b. Generalize part (a) and show that $$\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n} \leq \frac{x_{1}+\cdots+x_{n}}{n}$$

Describe the set of all points at which all three planes \(x+2 y+2 z=3, y+4 z=6,\) and \(x+2 y+8 z=9\) intersect.

Problems with two constraints Given a differentiable function \(w=f(x, y, z),\) the goal is to find its maximum and minimum values subject to the constraints \(g(x, y, z)=0\) and \(h(x, y, z)=0\) where \(g\) and \(h\) are also differentiable. a. Imagine a level surface of the function \(f\) and the constraint surfaces \(g(x, y, z)=0\) and \(h(x, y, z)=0 .\) Note that \(g\) and \(h\) intersect (in general) in a curve \(C\) on which maximum and minimum values of \(f\) must be found. Explain why \(\nabla g\) and \(\nabla h\) are orthogonal to their respective surfaces. b. Explain why \(\nabla f\) lies in the plane formed by \(\nabla g\) and \(\nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value. c. Explain why part (b) implies that \(\nabla f=\lambda \nabla g+\mu \nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value, where \(\lambda\) and \(\mu\) (the Lagrange multipliers) are real numbers. d. Conclude from part (c) that the equations that must be solved for maximum or minimum values of \(f\) subject to two constraints are \(\nabla f=\lambda \nabla g+\mu \nabla h, g(x, y, z)=0,\) and \(h(x, y, z)=0\)

Let \(E\) be the ellipsoid \(x^{2} / 9+y^{2} / 4+z^{2}=1, P\) be the plane \(z=A x+B y,\) and \(C\) be the intersection of \(E\) and \(P\). a. Is \(C\) an ellipse for all values of \(A\) and \(B\) ? Explain. b. Sketch and interpret the situation in which \(A=0\) and \(B \neq 0\). c. Find an equation of the projection of \(C\) on the \(x y\) -plane. d. Assume \(A=\frac{1}{6}\) and \(B=\frac{1}{2} .\) Find a parametric description of \(C\) as a curve in \(\mathbb{R}^{3}\). (Hint: Assume \(C\) is described by \(\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) and find \(a, b, c, d, e, \text { and } f .)\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.