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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 9-14, find all first and second partial derivatives.

z = y e x y 2

To determine

To calculate: The first and second partial derivatives of function z=yexy2.

Explanation

Given Information:

The provided function is z=yexy2.

Formula used:

The first partial derivatives of the function z=f(x,y) are zx=x[f(x,y)], zy=y[f(x,y)].

The second partial derivatives of the function z=f(x,y) are,

2fx2=x(fx)=fxx

And,

2fy2=y(fy)=fyy

Also,

2fxy=x(fy)=fyx

Also,

2fyx=y(fx)=fxy

Product rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then f(x)=u(x)v(x)+v(x)u(x).

Calculation:

Consider the function, z=yexy2

Use product rule,

The first partial derivatives of the function are,

zx=x(yexy2)=y(exy2y2)+exy20=y3exy2 …… (1)

And

zy=y(yexy2)=y(exy2x2y)+exy21=2xy2exy2+exy2 …… (2)

Differentiate (1) with respect to x,

2zx2=x(y3exy2)=y3(exy2y2)+ex

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