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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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In Exercises 20-23, find the first partial derivatives and evaluate each at the point ( 2 ,   3 ) .

f ( x , y ) = 3 x y 2 x + y

To determine

To calculate: The first partial derivatives for the function f(x,y)=3xy2x+y and their values at point (2,3).

Explanation

Given Information:

The provided function is f(x,y)=3xy2x+y and the point is (2,3).

Formula used:

If the function is f(x,y)=u(x,y)v(x,y),

Then, the first order partial derivative of f(x,y) with respect to x is,

fx(x,y)=v(x,y)ux(x,y)u(x,y)vx(x,y)[v(x,y)]2

The first order partial derivative of f(x,y) with respect to y is,

fy(x,y)=v(x,y)uy(x,y)u(x,y)vy(x,y)[v(x,y)]2

Calculation:

Consider the function,

f(x,y)=3xy2x+y

The first partial derivatives are,

fx=(x+y)(310)(3xy2)(1+0)(x+y)2=3x+3y3x+y2(x+y)2=y2+3y(x+y)2=y(3+y)(x+y)2

And,

fy(x,y)=(x+y)(02y)(3x

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