   Chapter 14.5, Problem 51E

Chapter
Section
Textbook Problem

Assume that all the given functions have continuous second-order partial derivatives.51. If z = f(x, y), where x = r2 + s2 and y = 2rs, find ∂2z/∂r ∂s. (Compare with Example 7.)

To determine

To find: The equation 2zrs if z=f(x,y) , where x=r2+s2andy=2rs .

Explanation

The given function is, z=f(x,y) .

Given that x=r2+s2 and y=2rs .

Find the partial derivative of xandy with respect to r ,

xr=x(r2+s2)=2ryr=r(2rs)=2s

Thus, the partial derivates, xr=2randyr=2s .

Find the partial derivative of xandy with respect to s ,

xs=y(r2+s2)=2sys=s(2rs)=2r

Thus, the partial derivates, uy=2sandvy=2r .

The partial derivative zs is computed as follows,

zs=zxxs+zyys=zx2s+zy2r

Then take partial derivative of   zs with respect to r and obtain the required value

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