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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 Partial Derivatives In Exercises 1-14, find the first partial derivatives. See

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

To determine

To calculate: The first partial derivatives for the function f(x,y)=xyx2+y2.

Explanation

Given information:

The provided function is f(x,y)=xyx2+y2.

Formula used:

Consider the function z=f(x,y) then for the value of zx consider y to be constant and differentiate with respect to x and the value of zy consider x to be constant and differentiate with respect to y.

Calculation:

Consider the provided function is,

f(x,y)=xyx2+y2

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

fx(x,y)=x(xyx2+y2)=(x2+y2)x(xy)(xy)x(x2+y2)(x2+y2)2=(x2+y2)(y)(xy)(2x)(x2+y2)2=(x2y+y3)(2x2y)(x2+y2)2

Further simplify the above equation.

fx(x,y)=x2y+y32x2y(x2+y2)2=x2y+y3(x2+y2)2

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

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