   Chapter 14.3, Problem 54E

Chapter
Section
Textbook Problem

Find all the second partial derivatives.54. f(x, y) = ln(ax + by)

To determine

To find: The second order partial derivative of the function f(x,y)=ln(ax+by) .

Explanation

The given function is, f(x,y)=ln(ax+by) .

Take the partial derivative of the given function with respect to x and obtain fx .

fx=1(ax+by)[a(1)+0]

fx=aax+by (1)

Take the partial derivative of the equation (1) with respect to x and fxx .

2fx2=a(ax+by)2[a(1)+0]=a2(ax+by)2=a2(ax+by)2

Thus, fxx(x,y)=a2(ax+by)2 .

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

fy=1(ax+by)[0+b(1)]

fy=bax+by (2)

Take the partial derivative of the equation (2) with respect to y and obtain fyy .

2fy2=b(ax+by)2[0+b(1)]=b2(ax+by)2=b2(ax+by)2

Hence, fyy(x,y)=b2(ax+by)2

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