   Chapter 7.4, Problem 16E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Finding Partial Derivatives In Exercises 15 and 16, use the limit definition of partial derivatives to find ∂ z / ∂ x  and  ∂ z / ∂ y . z   =   x 2  -  2 x y  +  y 2

To determine

To calculate: The value zx and zy for the function z=x22xy+y2 by using the limit definition.

Explanation

Given information:

The provided function is z=x22xy+y2.

Formula used:

According to the limit definition of the partial function, if z=f(x,y) then the derivative of the zx and zy is defined by,

zx=limΔx0f(x+Δx,y)f(x,y)Δxzy=limΔy0f(x,y+Δy)f(x,y)Δy

Calculation:

Consider the provided function is,

z=x22xy+y2

Partially derivative of the function z=x22xy+y2 with respect to x.

zx=limΔx0(x+Δx)22(x+Δx)y+y2(x22xy+y2)Δx=limΔx0x2+2xΔx+Δx22xy2yΔx+y2x2+2xyy2Δx=limΔx02xΔx+Δx22yΔxΔx=limΔx0(2x+Δx2y)

Further simplify the above equation

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