The partial derivative is a central concept in calculus, particularly in multivariable calculus. It measures the instantaneous rate of change of a function of several variables with respect to one chosen variable while the other variables are held fixed. This generalizes the ordinary derivative from single-variable calculus and underlies constructions such as the gradient, directional derivatives and many methods used in applied mathematics and the sciences.
Definition and notation
Let f be a function of two or more variables, for example f(x,y,...) . The partial derivative of f with respect to x at a point is defined by the limit ∂f/∂x = lim_{h→0} (f(x+h,y,...) − f(x,y,...)) / h, where all other coordinates are held constant. Common notations include ∂f/∂x, f_x, and ∂_x f. For the general notion of a derivative see derivative, and for discussion of dependent and independent symbols see variable. Further guidance on notation appears in many texts and online resources, for example notation guides.
Basic rules and higher-order derivatives
Partial derivatives obey many of the same algebraic rules as one-variable derivatives: linearity, product and quotient rules, and versions of the chain rule. When functions are composed, the multivariable chain rule expresses how partials combine according to how the inner variables depend on the outer parameters. Repeated differentiation yields higher-order partial derivatives such as ∂²f/∂x² and mixed partials like ∂²f/∂x∂y.
- Linearity: ∂/∂x (a f + b g) = a ∂f/∂x + b ∂g/∂x.
- Product rule: ∂/∂x (f g) = f_x g + f g_x.
- Chain rule (informal): if x and y depend on t, then df/dt = f_x dx/dt + f_y dy/dt.
Under suitable smoothness hypotheses (for instance, if the second partials are continuous in a neighborhood), mixed partials commute: ∂²f/∂x∂y = ∂²f/∂y∂x. This symmetry is commonly called Clairaut's theorem or Schwarz's theorem in different sources.
Examples and visualization
Consider f(x,y)=x² y + sin(xy). Differentiating with respect to x while holding y fixed gives ∂f/∂x = 2xy + y cos(xy), and with respect to y gives ∂f/∂y = x² + x cos(xy). Geometrically, a partial derivative is the slope of a curve obtained by slicing the graph of z=f(x,y) along a plane parallel to one coordinate axis; fixing y yields a curve in the x–z plane whose ordinary derivative equals ∂f/∂x. Such slices are a useful aid to intuition when visualizing surfaces and their tangent planes.
Gradient, Hessian and directional derivatives
The gradient vector of f is the tuple of its first partial derivatives and points in the direction of steepest increase. The Hessian matrix collects the second partial derivatives and is used to study curvature and classify critical points in optimization. A directional derivative generalizes the partial derivative: taken along any unit vector, it measures the rate of change in that direction; partials are the directional derivatives along coordinate basis directions.
Applications and cautions
Partial derivatives are fundamental in optimization (for locating extrema with constraints, via Lagrange multipliers), in formulating and solving partial differential equations that model physical phenomena (heat, wave and Laplace equations), and in economics for marginal analysis (how changing one input affects output holding others constant). When using partials one must be careful about which variables are considered independent, and about regularity conditions required to interchange differentiation and limits or to apply Taylor expansions reliably.
Further reading
Introductory and advanced accounts of partial derivatives can be found in standard calculus and analysis texts and in online course materials on multivariable methods and general calculus. For connections to related concepts see entries on notation, function theory and the general derivative.