Write a Chain Rule Formula for the Following Derivative A)

question 219

Multiple Choice

Write a chain rule formula for the following derivative.
- $$\frac { \partial \mathrm { w } } { \partial \mathrm { t } } \text { for } \mathrm { w } = \mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) ; \mathrm { x } = \mathrm { g } ( \mathrm { r } , \mathrm { s } , \mathrm { t } ) , \mathrm { y } = \mathrm { h } ( \mathrm { r } , \mathrm { s } , \mathrm { t } ) , \mathrm { z } = \mathrm { k } ( \mathrm { r } , \mathrm { s } , \mathrm { t } )$$

• A) \(\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { x } } \frac { \partial \mathrm { x } } { \partial \mathrm { t } } + \frac { \partial \mathrm { w } } { \partial \mathrm { y } } \frac { \partial \mathrm { y } } { \partial \mathrm { t } }\)
• B) \(\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { x } } \frac { \partial \mathrm { x } } { \partial \mathrm { r } } + \frac { \partial \mathrm { w } } { \partial \mathrm { y } } \frac { \partial \mathrm { y } } { \partial \mathrm { s } } + \frac { \partial \mathrm { w } } { \partial \mathrm { z } } \frac { \partial \mathrm { z } } { \partial \mathrm { t } }\)
• C) \(\frac { \partial w } { \partial t } = \frac { \partial w } { \partial x } + \frac { \partial w } { \partial y } + \frac { \partial w } { \partial z }\)
• D) \(\frac { \partial \mathrm { w } } { \partial \mathrm { t } } = \frac { \partial \mathrm { w } } { \partial \mathrm { x } } \frac { \partial \mathrm { x } } { \partial \mathrm { t } } + \frac { \partial \mathrm { w } } { \partial \mathrm { y } } \frac { \partial \mathrm { y } } { \partial \mathrm { t } } + \frac { \partial \mathrm { w } } { \partial \mathrm { z } } \frac { \partial \mathrm { z } } { \partial \mathrm { t } }\)

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Definitions:

Carboxylic Acid

Organic acids containing a carboxyl group (COOH), which makes them capable of donating a proton (acidic) and forming covalent bonds with metals.

Alcohol

An organic compound characterized by one or more hydroxyl (–OH) groups attached to a carbon atom, used in a wide array of applications from solvents to beverages.

IR Spectroscopy

A technique used to identify molecular structures through the absorption of infrared light, which causes molecular vibrations.

Compounds

Substances formed from the chemical combination of two or more elements, characterized by specific molecular compositions and properties.