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Partial Derivatives

Multivariable rates of change, gradients, Jacobians, and the chain rule in higher dimensions.

What Are Partial Derivatives?

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A partial derivative measures the rate of change of a multivariable function with respect to one variable while holding all other variables constant. The notation uses a curly d (the symbol for partial differentiation) rather than the straight d used for ordinary derivatives. This visual distinction immediately signals that the function depends on multiple variables. When computing a partial derivative, treat every variable except the one of interest as if it were an ordinary constant. For example, to find the partial derivative of x squared times y cubed with respect to x, treat y as constant: the result is two x times y cubed.
Definition of a partial derivative. Only x changes; y is held fixed. The symbol ∂ (curly d) indicates partial differentiation.

The Gradient Vector

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The gradient, denoted by the symbol nabla (an upside-down triangle), collects all the partial derivatives of a function into a single vector. This vector has a powerful geometric interpretation: it points in the direction of steepest ascent. If you are standing on a mountain, the gradient points straight uphill. The negative of the gradient points straight downhill — this is the basis of gradient descent, the optimization algorithm that trains neural networks. The magnitude of the gradient tells you how steep the slope is at that point.

Directional Derivatives

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The directional derivative generalizes partial derivatives to any direction. While a partial derivative measures the rate of change along a coordinate axis, a directional derivative measures the rate of change along any arbitrary direction specified by a unit vector u. The directional derivative equals the dot product of the gradient with the direction vector. This means the maximum rate of change occurs in the direction of the gradient, and the rate of change is zero in directions perpendicular to the gradient (level curves).
Directional derivative. The rate of change in direction u equals the dot product of the gradient with the unit direction vector.

The Jacobian Matrix

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The Jacobian is a matrix of FIRST-order partial derivatives for a vector-valued function. If a function maps from n-dimensional space to m-dimensional space, the Jacobian is an m by n matrix. The Jacobian determinant (for square Jacobians) measures how much the function stretches or compresses volumes locally. It is essential for the change of variables formula in multiple integrals. The chain rule for multivariable functions involves Jacobian matrices: the derivative of a composition equals the matrix product of the Jacobians.

Worked Example

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Worked Example
Find the gradient of f(x,y) = x²y + sin(x) at the point (π, 1).
Partial derivative with respect to x: 2xy + cos(x). At (π,1): 2π(1) + cos(π) = 2π − 1.
Partial derivative with respect to y: x². At (π,1): π².
So the gradient at (π, 1) is (2π − 1, π²).

Gradient Field Visualizer

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See how the gradient points uphill on a surface. Adjust the function parameters to explore different landscapes.