Definition

Noun: A mathematical concept representing the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is a fundamental tool in multivariable calculus.

Usage

The term "partial derivative" is used to analyze how a multivariable function changes in the direction of one specific variable. It is denoted by the symbol ∂ (the partial derivative symbol).

Examples

• To find the slope of a surface in the x-direction, you compute the partial derivative with respect to x.
• In thermodynamics, the partial derivative of internal energy with respect to volume at constant temperature is an important quantity.
• The function f(x, y) = x²y has a partial derivative with respect to x of 2xy.

Advanced Usage

• Notation: The partial derivative of a function with respect to is written as ∂f/∂x, fₓ, or Dₓf.
• Gradient Vector: The vector formed by all first-order partial derivatives of a function is called its gradient (∇f).
• Higher-Order Derivatives: One can take partial derivatives of partial derivatives, leading to concepts like the second partial derivative (e.g., ∂²f/∂x²) and mixed partial derivative (e.g., ∂²f/∂x∂y).

Variants and Related Words

• Partial Differentiation (n): The process or operation of calculating a partial derivative.
• Derivative (n): The more general concept of an instantaneous rate of change, of which a partial derivative is a specific type for multivariable functions.

Synonyms

• Partial differential coefficient (formal/technical)

Related Phrases

• Take the partial derivative of: To perform the calculation.
  • We need to take the partial derivative of the pressure function with respect to temperature.
• With respect to (w.r.t.): A standard prepositional phrase used in the notation and description.
  • Calculate the partial derivative with respect to time.

Noun

1. the derivative of a function of two or more variables with respect to a single variable while the other variables are considered to be constant