Partial derivative

A partial derivative measures how a multivariable function changes with respect to one variable while the other variables are held constant.

Limit of a function: A value a function approaches as its input approaches a specified point. The derivative is defined by taking a limit of difference quotients.
Multivariable function: A function whose input consists of two or more variables. Partial derivatives describe how such a function changes one input at a time.
Constrained optimization: The search for extrema of an objective function subject to specified restrictions. Partial derivatives help identify stationary points and enforce constraints.
Ordinary derivative: The derivative of a function of one variable with respect to that variable. It varies the sole input rather than selecting one among several.
Difference quotient: A ratio measuring a function's change over a finite change in its input. A partial derivative takes the limit of this ratio along one coordinate direction.
Independent variable: A variable whose value can be chosen as an input to a function or model. The selected independent variable is the one varied in a partial derivative.
Lagrange multiplier: A scalar used to locate constrained extrema by relating objective and constraint gradients. The method turns partial-derivative conditions into equations for constrained optima.
Total derivative: The derivative accounting for changes in all dependent variables as an input changes. Unlike a partial derivative, it includes indirect changes through dependent variables.
Gradient: The vector of a scalar-valued function's first partial derivatives. Its components collect all first-order rates of change at a point.
Coordinate system: A framework that assigns numerical coordinates to points in a space. Holding other coordinates fixed defines the direction used for a coordinate partial.

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