Chapter 14.2, Problem 42E

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Chapter
Section

### Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

# 42.  If   z = x 3 − 4 x 2 y + 5 y 3 ,  find  z y y x .

To determine

To calculate: The third partial derivative zyyx of the function z=x34x2y+5y3.

Explanation

Given Information:

The provided function is z=x3âˆ’4x2y+5y3.

Formula used:

For a function z(x,y), the third partial derivative, when first derivative is taken with respect to y and second derivative is taken with respect to y and third derivative is taken with respect to x is zyyx=âˆ‚3zâˆ‚xâˆ‚yâˆ‚y=âˆ‚âˆ‚x(âˆ‚âˆ‚y(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.

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Calculation:

Consider the function, z=x3âˆ’4x2y+5y3

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