A partial derivative is the derivative of a function with several variables, taken with respect to just one variable while keeping all the others fixed.

Intuition (quick picture in words)

Imagine a surface z=f(x,y)z=f(x,y)z=f(x,y) like a hilly landscape.

  • ∂f∂x\frac{\partial f}{\partial x}∂x∂f​ tells you how steeply the height zzz changes if you walk only in the xxx-direction, holding yyy fixed (like walking east–west on a specific line).
  • ∂f∂y\frac{\partial f}{\partial y}∂y∂f​ tells you how steeply the height changes if you walk only in the yyy-direction, holding xxx fixed (north–south on a specific line).

Formally, for a function f(x,y,… )f(x,y,\dots)f(x,y,…), a partial derivative with respect to xxx is just the usual derivative, but you treat every other variable as a constant. This idea is heavily used in multivariable calculus, optimization, and physics to describe how changing one input affects the output while everything else stays the same.

Example: For f(x,y)=x2yf(x,y)=x^{2}yf(x,y)=x2y,

  • ∂f∂x=2xy\frac{\partial f}{\partial x}=2xy∂x∂f​=2xy (treat yyy as a constant),
  • ∂f∂y=x2\frac{\partial f}{\partial y}=x^{2}∂y∂f​=x2 (treat xxx as a constant).

TL;DR: A partial derivative measures the rate of change of a multivariable function with respect to one variable at a time, keeping the others constant.