Chapter 14, Problem 18RE

### Mathematical Applications for the ...

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

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=âˆ‚2zâˆ‚x2=âˆ‚âˆ‚x(âˆ‚zâˆ‚x).

When both derivatives are taken with respect to y is zyy=âˆ‚2zâˆ‚y2=âˆ‚âˆ‚y(âˆ‚zâˆ‚y).

When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=âˆ‚2zâˆ‚yâˆ‚x=âˆ‚âˆ‚y(âˆ‚zâˆ‚x).

When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=âˆ‚2zâˆ‚xâˆ‚y=âˆ‚âˆ‚x(âˆ‚zâˆ‚y).

Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1.

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

Quotient rule for function f(x)=u(x)v(x), where u and v are differentiable functions of x, then fâ€²(x)=v(x)â‹…uâ€²(x)âˆ’u(x)â‹…vâ€²(x)(v(x))2.

Chain rule for function f(x)=u(v(x)) is fâ€²(x)=uâ€²(v(x))â‹…vâ€²(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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