The ordinary derivative of a function of two or more variables with respect to one of the variables, considering the others as constants.

Geometrically, the partial derivatives are equal to the slopes of the tangents to the curves which are the intersections of the surface z=f(x,y) and the planes whose equations are y=b and x=a, respectively.

For a function f:U→R (U⊆Rn some open set) and a point x=(x1,...,xn), define

fi,x(t) = f(x1,...,xi-1,t,xi+1,...,xn).
Then fi,x is a function of one variable when t is near xi, and we can do calculus with it. The partial derivative of f with respect to xi is defined to be
f/∂xi = dfi,x/dt.
Note the unfortunate double usage of x and especially xi to denote both a vector or a scalar variable and a particular value of that variable. Unfortunately, this practice is so common that I feel I must abide by it. This is a function ∂f/∂xi : U→R. If the derivative on the RHS does not exist, the partial derivative on the LHS doesn't, either. By tradition, the same letters are re-used, instead of new ones like t. Your multivariable calculus textbook may well follow this "convention", and give a seemingly different definition. The only purpose of this is to make a confusing subject more confusing.

Another notation for ∂f/∂xi is fxi. This notation is particularly useful for taking multiple partial derivatives: for a function f(x,y),

fxx = ∂2f/∂x2 = ∂/∂x(∂f/∂x);
fxy = ∂2f/∂y∂x = ∂/∂y(∂f/∂x);
fyx = ∂2f/∂x∂y = ∂/∂x(∂f/∂y);
fyy = ∂2f/∂y2 = ∂/∂y(∂f/∂y).
In particular, the Laplacian of f is Δf = fxx+fyy.

WHEN f is differentiable, it turns out that

f = (∂f/∂x1,...,∂f/∂xn).
But even if all partial derivatives exist at x, f might not be differentiable there -- be careful!

Contrary to what you might expect (or even read about in various places, once upon a time even on E2...), it is not necessarily true that ∂2f/∂xi∂xj = ∂2f/∂xj∂xi. You cannot, in general, interchange the order of the derivatives.

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