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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Second Partial Derivatives In Exercises 55–60, find the second partial derivatives.

f ( x , y ) = y x y

To determine

To calculate: The second partial derivative of function f(x,y)=yxy.

Explanation

Given Information:

The provided function is f(x,y)=yxy.

Formula used:

The power rule of differentiation with respect to x by holding y constant,

x[f(x,y)]n=n[f(x,y)]n1xf(x,y)

The power rule of differentiation with respect to y by holding x constant,

y[f(x,y)]n=n[f(x,y)]n1yf(x,y)

The quotient rule of differentiation with respect to x by holding y constant,

x[f(x,y)g(x,y)]=[g(x,y)xf(x,y)f(x,y)xg(x,y)][g(x,y)]2

The quotient rule of differentiation with respect to y by holding x constant,

y[f(x,y)g(x,y)]=[g(x,y)yf(x,y)f(x,y)yg(x,y)][g(x,y)]2

Calculation:

Consider the function,

f(x,y)=yxy

Differentiate with respect to x by holding y constant,

xf(x,y)=x(yxy)=(xy)x(y)(y)x(xy)(xy)2=(xy)(0)(y)(1)(xy)2=y(xy)2

The first partial derivative of function f(x,y)=yxy with respect to x,

xf(x,y)=y(xy)2

Now again differentiate with respect to x by holding y constant,

22xf(x,y)=x[y(xy)2]=(xy)2x(y)yx(xy)2(xy)2=(xy)2(0)2y(xy)(xy)4=2y(xy)3

Differentiate with respect to y by holding x constant,

yf(x,y)=y(yxy)=(xy)y(y)+(y)y(xy)(xy)2=(xy)(1)(y)(1)(xy)

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