Answered: In Problem find the second partials.…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.10: Partial Fractions

Problem 15E

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Question

In Problem find the second partials.

(a) z_xx (b) z_yy (c) z_xy (d) z_yx

Expert Solution

Step 1

Given function is $z = x^{2} e^{y^{2}}$ .

To find:

$(a) z_{x x}$ (b) $z_{y y}$ (c). $z_{x y}$ (d). $z_{y x}$

Solution:

Partial derivative of a multi-variables function is derivative of function with respect to one variable, keeping other variable as constant.

Finding partial derivative of function with respect to $x$ , keeping y as constant.

$\begin{array}{rcl} z_{x} & = & \frac{\partial}{\partial x} [x^{2} e^{y^{2}}] \\ = & e^{y^{2}} \cdot \frac{\partial}{\partial x} (x^{2}) \\ = & e^{y^{2}} \cdot 2 x \\ = & 2 x e^{y^{2}} \end{array}$

Step 2

Now, finding $z_{x x}$ (derivative of $z_{x}$ with respect to $x$ ).

$\begin{array}{rcl} z_{x x} & = & \frac{\partial}{\partial x} (z_{x}) \\ = & \frac{\partial}{\partial x} (2 x e^{y^{2}}) \\ = & 2 e^{y^{2}} \cdot \frac{\partial}{\partial x} (x) \\ = & 2 e^{y^{2}} \cdot 1 \\ = & 2 e^{y^{2}} \end{array}$

Now, to find $z_{x y}$ , we need to find partial derivative of $z_{x}$ with respect to y.

$\begin{array}{rcl} z_{x y} & = & \frac{\partial}{\partial y} (z_{x}) \\ = & \frac{\partial}{\partial y} (2 x e^{y^{2}}) \\ = & 2 x \cdot e^{y^{2}} \cdot 2 y \\ = & 4 x y e^{y^{2}} \end{array}$

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