   Chapter 14.4, Problem 15E

Chapter
Section
Textbook Problem

Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.15. f(x, y) = 4arctan(xy), (1, 1)

To determine

To find: The linearization L(x,y) of the function f(x,y)=4arctan(xy) at the point (1,1).

Explanation

Theorem used:

“If the partial derivatives fx and fy exist near (a,b) and are continuous at (a,b) , then f is differentiable at (a,b) .”

Result used:

“Linearization of the function f(x,y) at (a,b) is L(x,y)=f(a,b)+fx(a,b)(xa)+fy(a,b)(yb) .”

Calculation:

Let the given function as z=4tan1(xy) . (1)

The point is P(a,b)=(1,1) .

Take partial derivative with respect to x in the equation (1),

fx(x,y)=411+(xy)2(y)[byx(tan1(x))=11+x2]=4y1+(xy)2

Take partial derivative with respect to y in the equation (1),

fy(x,y)=411+(xy)2(x)[byx(tan1(x))=11+x2]=4x1+(xy)2

Obtain the partial derivative of x at the given point (1,1).

fx(a,b)=fx(1,1)=4(1)1+(1(1))2=42=2

Obtain the partial derivative of y at the given point (1,1)

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