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A partial derivative measures how a multivariable function changes when one variable changes while all other variables are held constant. If f(x, y) is a function of two variables, the partial derivative with respect to x, written as a curly-d f / curly-d x, tells you the instantaneous rate of change of f in the x-direction at any given point. The formal definition mirrors the ordinary derivative but fixes all variables except the one being differentiated: curly-d f / curly-d x = lim as h approaches 0 of [f(x+h, y) - f(x, y)] / h.

Several notations are used for partial derivatives, and understanding them is essential when reading textbooks and research papers. The most common is Leibniz notation: curly-d f / curly-d x. Subscript notation writes the same thing as f_x or f_y. The operator notation D_x f is also used, particularly in differential equations. For higher-order derivatives, curly-d squared f / curly-d x squared denotes the second partial with respect to x, while curly-d squared f / (curly-d x curly-d y) denotes a mixed partial derivative.

Geometrically, a partial derivative has a clear visual interpretation. For a function f(x, y) that defines a surface in three-dimensional space, the partial derivative curly-d f / curly-d x at a point (a, b) gives the slope of the tangent line to the curve formed by slicing the surface with the plane y = b. In other words, you freeze y at the value b and look at how the surface rises or falls as x changes. Similarly, curly-d f / curly-d y gives the slope of the tangent line when you slice with the plane x = a. These tangent lines together define the tangent plane to the surface at that point.

The gradient vector, denoted nabla f or grad f, collects all partial derivatives into a single vector: nabla f = (curly-d f / curly-d x, curly-d f / curly-d y) in two dimensions, or (curly-d f / curly-d x, curly-d f / curly-d y, curly-d f / curly-d z) in three dimensions. The gradient always points in the direction of steepest ascent of the function and its magnitude equals the maximum rate of change. This makes it central to optimisation. A directional derivative in any unit vector direction u can be computed as the dot product of the gradient with u: D_u f = nabla f dot u. Level curves (contours) of f are always perpendicular to the gradient.

Higher-order partial derivatives are obtained by differentiating a partial derivative again. The second-order partial curly-d squared f / curly-d x squared measures the concavity of f in the x-direction. Mixed partial derivatives, such as curly-d squared f / (curly-d x curly-d y), measure how the rate of change in one direction varies as the other variable changes. Clairaut's theorem (also called Schwarz's theorem) states that if the mixed partial derivatives are continuous, then the order of differentiation does not matter: curly-d squared f / (curly-d x curly-d y) = curly-d squared f / (curly-d y curly-d x). The Hessian matrix collects all second-order partial derivatives and is used to classify critical points as local maxima, local minima, or saddle points.

Partial derivatives have wide-ranging applications across science, engineering, economics, and computing. In physics, thermodynamic quantities such as entropy, pressure, and temperature are related through partial derivatives (Maxwell relations). In electromagnetism, Maxwell's equations involve partial derivatives of electric and magnetic fields. In economics, marginal cost is the partial derivative of total cost with respect to quantity, while marginal utility is the partial derivative of a utility function with respect to the quantity of a good consumed. In machine learning and artificial intelligence, gradient descent is the primary optimisation algorithm: it computes the gradient (vector of partial derivatives) of a loss function with respect to model parameters and updates those parameters in the direction of steepest descent. Backpropagation in neural networks is fundamentally a chain of partial derivative calculations using the multivariable chain rule.