Answer all parts a) Find all second-order partial derivatives of the function f(x,y) = ycos(x²y).dwdwb)Find aswhere w ==, x=s+t, y=st, and z = s – t.anddty+z

Name: Answer all parts a) Find all second-order partial derivatives of the f
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Description: Answer all parts a) Find all second-order partial derivatives of the function f(x,y) = ycos(x²y).dwdwb)Find aswhere w ==, x=s+t, y=st, and z = s – t.anddty+z

Answered: a) Find all second-order partial…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...

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Answer all parts

a) Find all second-order partial derivatives of the function f(x,y) = ycos(x²y).
dw
dw
b)Find as
where w ==, x=s+t, y=st, and z = s – t.
and
dt
y+z

Expert Solution

Step 1: (a).

Given, the function $f (x, y) = y \cos (x^{2} y) .$ we have to find all the second-order partial derivatives of the function.

$\begin{array}{rcl} f (x, y) & = & y \cos (x^{2} y) \\ \frac{\partial f}{\partial x} & = & y [- \sin (x^{2} y) \cdot 2 x y] = - 2 y^{2} \cdot x \sin (x^{2} y) \\ \frac{\partial^{2} f}{\partial x^{2}} & = & - 2 y^{2} \cdot \frac{\partial}{\partial x} [x \sin (x^{2} y)] \\ \frac{\partial^{2} f}{\partial x^{2}} & = & - 2 y^{2} \cdot [\frac{\partial}{\partial x} [x] \cdot \sin (x^{2} y) + x \cdot \frac{\partial}{\partial x} \sin (x^{2} y)] \\ \frac{\partial^{2} f}{\partial x^{2}} & = & - 2 y^{2} \cdot [1 \cdot \sin (x^{2} y) + x \cdot - \cos (x^{2} y) \cdot 2 x y] \\ \frac{\partial^{2} f}{\partial x^{2}} = - 2 y^{2} [\sin (x^{2} y) - 2 x^{2} y \cos (x^{2} y)] \\ \frac{\partial f}{\partial y} & = & \frac{\partial}{\partial y} [y \cos (x^{2} y)] = \frac{\partial}{\partial y} [x] \cdot \cos (x^{2} y) + x \cdot \frac{\partial}{\partial y} [\cos (x^{2} y)] \\ \frac{\partial f}{\partial y} & = & \cos (x^{2} y) + x \cdot - \sin (x^{2} y) \cdot x^{2} \\ \frac{\partial f}{\partial y} & = & \cos (x^{2} y) - x^{3} \sin (x^{2} y) \\ \frac{\partial^{2} f}{\partial y^{2}} & = & \frac{\partial}{\partial y} [\cos (x^{2} y) - x^{3} \sin (x^{2} y)] \\ \frac{\partial^{2} f}{\partial y^{2}} & = & - \sin (x^{2} y) \cdot x^{2} - x^{3} \cos (x^{2} y) \cdot x^{2} \\ \frac{\partial^{2} f}{\partial y^{2}} = - x^{2} [\sin (x^{2} y) - x^{3} \cos (x^{2} y)] \end{array}$

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