Find the Indicated Partial Derivative $$u = x ^ { a } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$$

question 38

Multiple Choice

Find the indicated partial derivative.
$$u = x ^ { a } y ^ { b } z ^ { c } ; \frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } , a > 1 , b > 2 , c > 3$$

• A)
• \(\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = c b ( b - 1 ) c ( a - 1 ) ( a - 2 ) x ^ { c - 1 } y ^ { b - 2 } z ^ { a - 3 }\)
• B) \(\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { a - 1 } y ^ { 3 - 2 } z ^ { c - 3 }\)
• C)
• \(\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a b ( b - 1 ) c ( c - 1 ) ( c - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }\)
• D)
• \(\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = a c b ( a - 1 ) ( a - 2 ) x ^ { a - 1 } y ^ { b - 2 } z ^ { c - 3 }\)
• E)
• \(\frac { \partial ^ { 6 } u } { \partial x \partial y ^ { 2 } \partial z ^ { 3 } } = x ^ { b - 1 } y ^ { c - 2 } z ^ { a - 3 }\)

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Definitions:

Acid Chloride

A functional group in organic chemistry, characterized by a carbon atom double-bonded to an oxygen atom and bonded to a chlorine atom.

Primary Alcohol

An alcohol in which the hydroxyl group (-OH) is attached to a carbon atom that has only one other alkyl group linked to it.

Butyllithium

A lithium organic compound often used as a strong base in the synthesis of organic compounds.

Reaction

The process by which two or more reactants form one or more products, often releasing or absorbing energy.