Partial derivative

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Definition at a point

Generic definition

Suppose f is a function of more than one variable, where x is one of the input variables to f. Fix a choice x = x_0 and fix the values of all the other variables. The partial derivative of f with respect to x, denoted \partial f/\partial x, or f_x, is defined as the derivative at x_0 of the function that sends x to f at x for the same fixed choice of the other input variables.

For a function of two variables

Suppose f is a real-valued function of two variables x,y, i.e., the domain of f is a subset of \R^2. We define the partial derivatives as follows:

  • Partial derivative with respect to x:

\frac{\partial f(x,y)}{\partial x}|_{(x,y) = (x_0,y_0)} = \frac{d}{dx}f(x,y_0)|_{x = x_0}

In words, it is the derivative at x = x_0 of the function x \mapsto f(x,y_0).

This partial derivative is also denoted f_x(x_0,y_0) or f_1(x_0,y_0).

  • Partial derivative with respect to y:

\frac{\partial f(x,y)}{\partial y}|_{(x,y) = (x_0,y_0)} = \frac{d}{dy}f(x_0,y)|_{y = y_0}

In words, it is the derivative at y = y_0 of the function y \mapsto f(x_0,y).

This partial derivative is also denoted f_y(x_0,y_0) or f_2(x_0,y_0).

For a function of multiple variables

The notation here gets a little messy, so read it carefully. We consider a function f of n variables, which we generically denote (x_1,x_2,\dots,x_n) respectively. Consider a point Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): (a_1_a,2,\dots,a_n) in the domain of the function. In other words, this is a point where x_1 = a_1,x_2 =a_2, \dots, x_n = a_n.

Suppose i is a natural number in the set \{ 1,2,3,\dots,n \}.

The partial derivative at this point (a_1,a_2,\dots,a_n) with respect to the variable x_i is defined as the derivative:

\frac{d}{dx_i}f(a_1,a_2,\dots,a_{i-1},x_i,a_{i+1}, \dots,a_n)|_{x_i = a_i}

In other words, it is the derivative (evaluated at a_i) of the function x \mapsto f(x_1,x_2,\dots,x_{i-1},a_i,x_{i+1},\dots,x_n) with respect to x_i, evaluated at the point x_i = a_i.