hungryhippo
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Let's say I have a function where x, y, and z are all in feet. Taking the partial derivatives of x and y (dz/dx and dz/dy); what units would the partial derivatives be in?
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units of partial derivatives
- Thread starter hungryhippo
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dx and x will have the same unit (feet in this case).Let's say I have a function where x, y, and z are all in feet. Taking the partial derivatives of x and y (dz/dx and dz/dy); what units would the partial derivatives be in?
What will be the unit of dy?
What will be the unit of dy/dx?
HallsofIvy
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"dy/dx" is the instantaneous rate of change of y relative to the instantaneous rate of change of x. Ignoring the "instantaneous" for the moment, suppose that, in the time x changes from 3 feet to 6 feet, y changes from 4 feet to 16 feet. y has changed by 16- 4=12 feet while x has changed by 6- 3= 3 feet. y has changed (12 feet)/(3 feet)= 4, 4 times as fast as x. "four times as fast" has no units!
On the other hand, suppose that, as time, t, changes from 0 seconds to 2 seconds, x changes from 3 feet to 12 feet. x has changed by 12- 3= 9 feet while t has changed by 2 seconds. x has changed, relative to t, by (9 feet)/(2 seconds)= 4.5 feet/seconds= 4.5 "feet per second".
While the derivative is not a fraction, it is the limit of a fraction and can be treated like a fraction. The units of dy/dx are whatever units y has over what ever units x has. If y is the temperature of water, in degrees Celcius and t is time in hours, then dy/dt has units of "degrees Celcius per hour".
On the other hand, suppose that, as time, t, changes from 0 seconds to 2 seconds, x changes from 3 feet to 12 feet. x has changed by 12- 3= 9 feet while t has changed by 2 seconds. x has changed, relative to t, by (9 feet)/(2 seconds)= 4.5 feet/seconds= 4.5 "feet per second".
While the derivative is not a fraction, it is the limit of a fraction and can be treated like a fraction. The units of dy/dx are whatever units y has over what ever units x has. If y is the temperature of water, in degrees Celcius and t is time in hours, then dy/dt has units of "degrees Celcius per hour".