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Chapter 12: Q 47. (page 944)

In Exercises 43-50, compute all of the second-order partial derivatives for the functions fx,y=xsinyand show that the mixed partial derivatives are equal.

Short Answer

Expert verified

The second order partial derivatives for the function are

∂2f∂x2=0∂2f∂y2=-xsiny

Step by step solution

01

Step 1. Definition 

Clairaut's theorem on equality of mixed partials states that under assumption of continuity of both the second-order mixed partials of a function of two variables, the two mixed partials are equal.

02

Step 2. Finding second order partial derivative 

fx,y=xsiny⋯⋯1
Partially differentiate equation 1both sides with respect tox

⇒∂f∂x=siny

Again partially differentiate both sides with respect tox

⇒∂2f∂x2=0

Partially differentiate equation 1both sides with respect toy

⇒∂f∂y=xcosy

Again partially differentiate both sides with respect to y

⇒∂2f∂y2=-xsiny

03

Step 3. Finding mixed order partial derivative  

fx,y=xsiny

⇒∂2f∂x∂y=∂∂xxcosy⇒∂2f∂x∂y=cosy

Also

⇒∂2f∂y∂x=∂∂ysiny⇒∂2f∂y∂x=cosy

Now as we observe

∂2f∂x∂y=∂2f∂y∂x[Hence proved]

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