Partial derivatives freeze every input but one
A partial derivative differentiates a multivariable function with respect to one variable while holding the others fixed—unlike a total derivative that lets all inputs move. The idea powers gradients, differentials, and PDE formulations. Its curly ∂ symbol goes back to Condorcet in 1770, and Jacobi revived it in 1841.
A partial derivative differentiates a function of several variables with respect to one of them while the rest are held constant. Like an ordinary derivative it is defined as a limit: nudge only the i-th input by h, divide the change in output by h, and let h shrink to zero. That makes it the directional derivative taken along the unit vector of that variable.
The curly symbol ∂ first appeared with Marquis de Condorcet in 1770, who used it for partial differences. Adrien-Marie Legendre created the modern partial derivative notation in 1786 but later dropped it, and Carl Gustav Jacob Jacobi brought the symbol back in 1841. Alternatives include subscript forms such as f′ₓ and ∂ₓf, and Euler's operator notation is handier when a partial derivative must be evaluated at an awkward point such as (17, u + v, v²), where Leibniz notation becomes clumsy.
Having every partial derivative at a point does not guarantee the function is continuous there. If all partial derivatives exist near the point and are continuous, though, the function is totally differentiable with a continuous total derivative, and is called C1. A partial derivative is itself a function that can be differentiated again; when the second differentiation is in a different direction the result is a mixed partial derivative.
When every second-order mixed partial is continuous, the function earns the label C2 and the order of differentiation can be swapped. Where variables are linked to one another, fields such as statistical mechanics state explicitly which ones are held constant, for instance writing the derivative of f with respect to x with y and z fixed.
Source: Partial derivative
