   Chapter 7.4, Problem 7CP ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Find the second partial derivatives of f ( x ,   y ,   z ) = x e y +   2 x z   +   y 2 .

To determine

To calculate: The second partial derivatives of f(x,y,z)=xey+2xz+y2

Explanation

Given Information:

Function of three variable f(x,y,z)=xey+2xz+y2

Formula used:

For a multivariate function f(x,y,z) the first partial derivatives are:

fx=dfdx(y,zconstant)fy=dfdy(x,zconstant)fz=dfdz(x,yconstant)

The second partial derivatives are

fxx=d2fdx2fyy=d2fdy2fzz=d2fdz2fxy=d2fdxdy

And

fyx=d2fdydxfxz=d2fdxdzfzx=d2fdzdxfyz=d2fdydz

And

fxy=d2fdzdy

Calculation:

Consider the given equation

f(x,y,z)=xey+2xz+y2

Now, find the first partial derivative for above function

dfdx=ddx(xey+2xz+y2)=ey+2zdfdy=ddy(xey+2xz+y2)=xey+2ydfdz=ddz(xey+2xz+y2)=2x

And, second partial derivatives

fxx=d2fdx2=ddx(dfdx)=ddx(ey+2z)=0fyy=d2fdy2=ddy(dfdy)=ddy(xey+2y)=x<

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