Higher-order DerivativesFirst, we introduce some notation. If f be function of æ and y, thenafafandare functions of x and y also. Hence, we may have the following:second-order partial derivative:a (af= fzz - partial derivative of-with respect to r%3Da (af= fyv - partial derivative of-sewith respect to ydy%3Da (afdyðr dy dxa (afdrðy dr \dyse= fry - partial derivative ofwith respect to y%3Daf= fyz - partial derivative of-dywith respect to rthird-order partial derivatives:a faf (af|= frye - partial derivative of-wrt x--dzðyðr dr [dy (dr,dyðra faf= fygz – partial derivative of-дуwrt x%3DThe other third-order partial derivatives of functions in two variablesare also defined similarly. The above notations can be extended into functionsin n variables. 2. In each of the following, find fyzz, fæyx, fryz-2. f(x, y, z) = ln(xyz) + z² tan-1(e"y)

To find the partial derivatives of the given function use the differentiation rules to evaluate the…

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2. In each of the following, find fyzz, fryz, fryz- 2. f(x, y, z) = In(xyz) + z² tan-'(e*")

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter5: A Survey Of Other Common Functions

Section5.4: Combining And Decomposing Functions

Problem 14E: Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and...

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Question

Higher-order Derivatives
First, we introduce some notation. If f be function of æ and y, then
af
af
and
are functions of x and y also. Hence, we may have the following:
second-order partial derivative:
a (af
= fzz - partial derivative of-
with respect to r
%3D
a (af
= fyv - partial derivative of-
se
with respect to y
dy
%3D
a (af
dyðr dy dx
a (af
drðy dr \dy
se
= fry - partial derivative of
with respect to y
%3D
af
= fyz - partial derivative of-
dy
with respect to r
third-order partial derivatives:
a faf (af
|= frye - partial derivative of-
wrt x
--
dzðyðr dr [dy (dr,
dyðr
a faf
= fygz – partial derivative of-
ду
wrt x
%3D
The other third-order partial derivatives of functions in two variables
are also defined similarly. The above notations can be extended into functions
in n variables.

2. In each of the following, find fyzz, fæyx, fryz-
2. f(x, y, z) = ln(xyz) + z² tan-1(e"y)

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