   Chapter 14.3, Problem 71E

Chapter
Section
Textbook Problem

If f ( x , y , z )   = x y 2 z 3 +  arcsin ( x z ) , find fxzy. [Hint: Which order of differentiation is easiest?]

To determine

To find: The derivative fxzy of the function f(x,y,z)=xy2z3+arcsin(xz) .

Explanation

The given function is, f(x,y,z)=xy2z3+arcsin(xz) .

The third order partial derivative fxzy can be expressed fyxz as the third is continuous. So, first find the partial derivative of the given function with respect to y and hence the sin term will get eliminated.

Take the partial derivative of the given function with respect to y and obtain fy .

fy=y(xy2z3+arcsin(xz))=y(xy2z3)+y(arcsin(xz))=xz3y(y2)+y(arcsin(xz))=xz3(2y)+0=2xyz3

Thus, fy=2xyz3 . (1)

Take the partial derivative of the equation (1) with respect to z and obtain fyx

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