Partial derivative

The derivative with respect to one coordinate of a function f:ℝ^k→ℝ^m.

Partial derivative

Let $U\subseteq\mathbb{R}^k$ be open , let $f:U\to\mathbb{R}$ be a scalar-valued function, and let $a=(a_1,\dots,a_k)\in U$ . The partial derivative of $f$ with respect to the $j$ th variable at $a$ is

\frac{\partial f}{\partial x_j}(a) :=\lim_{h\to 0}\frac{f(a_1,\dots,a_j+h,\dots,a_k)-f(a_1,\dots,a_j,\dots,a_k)}{h},

provided the limit exists.

Partial derivatives measure the sensitivity of $f$ to changes in a single coordinate direction while holding all other coordinates fixed. Existence of partial derivatives alone does not imply differentiability of $f$ as a multivariable map.

Examples:

If $f(x,y)=x^2y$ , then $\frac{\partial f}{\partial x}(x,y)=2xy$ and $\frac{\partial f}{\partial y}(x,y)=x^2$ .
If $f(x,y)=x+y$ , then both partial derivatives are constantly $1$ .
There exist functions with all partial derivatives at a point but not continuous or not differentiable there (standard pathology examples).