Partial Derivative

analysis

Definition

Partial Derivative

Let $D\subseteq {\mathbb{R}}^n$ an open set, $f:D\to \mathbb{R}$ a function and $(x_1,x_2,\dots ,x_n)\in D$.

Then, the partial derivative of $f$ with respect to $x_i$ is given by:

$$\frac{\partial f}{\partial x_i}(x_1,x_2,\dots ,x_i,\dots x_n)=\underset{x\to x_i}{\lim }\frac{f(x_1,x_2,\dots ,x,\dots ,x_n)-f(x_1,x_2,\dots ,x_i,\dots ,x_n)}{x-x_i}$$