Find the Total Differential of the Function $$z = - \frac { 1 } { x ^ { 5 } + y ^ { 10 } }$$

question 124

Multiple Choice

Find the total differential of the function $$z = - \frac { 1 } { x ^ { 5 } + y ^ { 10 } }$$

• A)\(d z = \frac { x ^ { 6 } } { \left( x ^ { 5 } + y ^ { 10 } \right) ^ { 2 } } d x + \frac { y ^ { 11 } } { \left( x ^ { 5 } + y ^ { 10 } \right) ^ { 2 } } d y\)
• B)\(d z = \frac { x ^ { 6 } } { 6 \left( x ^ { 5 } + y ^ { 10 } \right) ^ { 2 } } d x + \frac { y ^ { 11 } } { 11 \left( x ^ { 5 } + y ^ { 10 } \right) ^ { 2 } } d y\)
• C)\(d z = \frac { 5 x ^ { 4 } } { x ^ { 5 } + y ^ { 10 } } d x + \frac { 10 y ^ { 9 } } { x ^ { 5 } + y ^ { 10 } } d y\)
• D)\(d z = \frac { 5 x ^ { 4 } } { \left( x ^ { 5 } + y ^ { 10 } \right) ^ { 2 } } d x + \frac { 10 y ^ { 9 } } { \left( x ^ { 5 } + y ^ { 10 } \right) ^ { 2 } } d y\)
• E)\(d z = \frac { x ^ { 6 } } { 6 \left( x ^ { 5 } + y ^ { 10 } \right) } d x + \frac { y ^ { 11 } } { 11 \left( x ^ { 5 } + y ^ { 10 } \right) } d y\)

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Definitions:

Break-even Point

The point at which total costs and total revenues are equal, meaning no net loss or gain is incurred.

Fixed Expenses

Costs that do not fluctuate with the level of production or sales, such as rent, insurance, and property taxes.

Margin of Safety

The difference between actual or projected sales and the break-even point, measured to assess the risk of loss.

Sales

The total revenue earned from the sale of goods or services over a specific period of time.