   Chapter 14, Problem 18RE Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

In Problems 15-18, find the second partials. ( a ) z x x     ( b ) z y y         ( c )   z x y                 ( d )   z y x 18.   z = ln ( x y + 1 )

(a)

To determine

To calculate: The second partial derivative zxx of the function z=ln(xy+1).

Explanation

Given Information:

The provided function is z=ln(xy+1).

Formula used:

For a function z(x,y), the second partial derivative,

When both derivatives are taken with respect to x is zxx=2zx2=x(zx).

When both derivatives are taken with respect to y is zyy=2zy2=y(zy).

When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=2zyx=y(zx).

When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=2zxy=x(zy).

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Derivative of natural logarithmic functions is such that, if y=lnu, where u is a differentiable function of x then dydx=1ududx.

Quotient rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then f(x)=v(x)u(x)u(x)v(x)(v(x))2.

Chain rule for function f(x)=u(v(x)) is f(x)=u(v(x))v(x)

(b)

To determine

To calculate: The second partial derivative zyy of the function z=ln(xy+1).

(c)

To determine

To calculate: The second partial derivative zxy of the function z=ln(xy+1).

(d)

To determine

To calculate: The second partial derivative zyx of the function z=ln(xy+1).

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