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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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In Exercises 12 and 13, find the second partial derivatives.

w = x 2 y 3 y 3 2 y z 2

To determine

To calculate: The second partial derivative of function ω=x2y3y32yz2.

Explanation

Given Information:

A polynomial (ω) function of three variables (x,y,z) is given as ω=x2y3y32yz2 .

Formula used:

First partial derivative of the function.

fx=ωxfy=ωyfz=ωz

Second partial derivatives of the function.

fxx=2ωx2fyy=2ωy2fzz=2ωz2

And,

fxy=2ωxyfxz=2ωxzfyz=2ωyz

Calculation:

Consider the given equation.

ω=x2y3y32yz2 …… (1)

Now, differentiate equation (1) partially with respect to x.

ωx=x(x2y3y32yz2)ωx=2xy …… (2)

Again, differentiating partially with respect to x.

2ωx2=x(2xy)2ωx2=2y

Differentiating equation (1) partially with respect to y .

ωy=y(x2y3y32yz2)ωy=x29y22z2 …… (3)

Again, differentiating above equation (3) with respect to y .

y(ωy)=y(x29y22z2)2ωy2=18y

Differentiating equation (1) partially with respect to z .

ωz=z(x2y3y32yz2)ωz=4yz …… (4)

Again, differentiating partially with respect to z.

2ωz2=4y

Differentiating equation (2) partially with respect to y

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