   Chapter 14.3, Problem 46E

Chapter
Section
Textbook Problem

Use the definition of partial derivatives as limits (4) to find fx(x, y) and fy(x, y).46. f ( x , y ) = x x + y 2

To determine

To find: The value of fx(x,y) and fy(x,y) by using the definition of partial derivatives as limits.

Explanation

Formula used:

If z=f(x,y) , then the partial derivative functions are,

fx(x,y)=fx=xf(x,y)fy(x,y)=fy=yf(x,y)

Definition used:

If f is a function of two variables, its partial derivatives fx and fy are defined as a limit by,

fx(x,y)=limh0f(x+h,y)f(x,y)hfy(x,y)=limh0f(x,y+h)f(x,y)h

Calculation:

The given function is, f(x,y)=xx+y2 (1)

Find the partial derivative of fx(x,y) by using the definition of partial derivative as limits.

fx(x,y)=limh0f(x+h,y)f(x,y)h=limh0x+hx+h+y2xx+y2h=limh0(x+h)(x+y2)x(x+h+y2)(x+h+y2)(x+y2)h=limh0(x+h)(x+y2)x(x+h+y2)h(x+h+y2)(x+y2)

Simplify further as follows,

fx(x,y)=limh0x2+xh+xy2+y2hx2xhxy2h(x+h+y2)(x+y2)=limh0y2hh(x+h+y2)(x+y2)=limh0y2(x+h+y2)(x+y2)

Apply the limit value and obtain the value of fx(x,y) .

fx(x,y)=y2(x+0+y2)(x+y2)=y2(x+y2)(x+y2)=y2(x+y2)2

Therefore, the partial derivative with respect to x is, fx(x,y)=y2(x+y2)2

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