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Partial derivatives
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Rate of change with respect to one variable holding others fixed.
∂f/∂xA partial derivative ∂f/∂x measures how f changes with x while all other variables are held fixed. It is used when a quantity depends on several variables, like temperature depending on position and time.
If f = x²y then ∂f/∂x = 2xy and ∂f/∂y = x².
Holding "the rest fixed" matters: ∂/∂x is not the same as the total derivative if other variables also change with x.
In more detail
A partial derivative measures how a function changes with one variable while the others are held fixed. Physics uses them when a quantity depends on several variables, such as pressure depending on both volume and temperature.
Real-world uses
- The heat capacity of a gas at constant volume is a partial derivative of internal energy with respect to temperature.
- Wave equations use partial derivatives in both space and time.
- The pressure in a gas depends on temperature and volume, and a partial derivative with respect to volume holds temperature fixed to find the change.
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Frequently asked questions
How does a partial derivative differ from an ordinary one?
An ordinary derivative has one variable. A partial derivative keeps all other variables constant while changing one.
Is the notation different?
Yes. Partial derivatives use the curly symbol instead of d, as in ∂f/∂x.
Why does the partial derivative symbol look different?
The curly d warns that other variables are being held constant, so the result depends only on the one variable you chose.
Related topics in Measurement & Math Tools
Triple products · Calculus: derivatives · Calculus: integrals · Gradient, divergence, curl · Differential equations in physics · Taylor series and small angles