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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 and Evaluating Partial Derivatives In Exercises 17-24, find the first partial derivatives and evaluate each at the given point. See Example 2.

f ( x ,   y )   =   4 x y x 2  +  y 2 ;   ( 1 ,  0 )

To determine

To calculate: The first partial derivatives of the function f(x,y)=4xyx2+y2 at the point (1,0).

Explanation

Given information:

The provided function is f(x,y)=4xyx2+y2 and the point is (1,0).

Formula used:

The first partial derivatives of z=f(x,y) are represented as,

zx=fx(x,y)=zx=x[fx(x,y)]zy=fy(x,y)=zy=y[fy(x,y)]

The value of first partial derivatives at point (a,b) are represented as,

zx|(a,b)=fx(a,b)zy|(a,b)=fy(a,b)

Calculation:

Consider the provided function is,

f(x,y)=4xyx2+y2

Partially derivative of the function f(x,y)=4xyx2+y2 with respect to x.

zx=fx(x,y)=x(4xyx2+y2)=(x2+y2)x(4xy)(4xy)x(x2+y2)(x2+y2)2=(x2+y2)(4y)(4xy)(12x2+y22x)(x2+y2)

Further simplify the above equation.

zx=(x2+y2)(4y)(4x2y)(x2+y2)32=4x2y+4y34x2y(x2+y2)32=4y3(x2+y2)32

Substitute (x,y)=(1,0) in derivative fx(x,y)

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