Partial Derivatives

The partial derivative of $f(x_1, \ldots, x_n)$ with respect to $x_i$ measures the rate of change of $f$ when $x_i$ changes and all other variables are held fixed. It is computed exactly like a one-variable derivative — treating every variable except $x_i$ as a constant.

For a function $f: \mathbb{R}^2 \to \mathbb{R}$, the partial derivative with respect to $x$ at point $(a, b)$ is: $\frac{\partial f}{\partial x}(a, b) = \lim_{h \to 0} \frac{f(a+h, b) - f(a, b)}{h}$ This is simply the ordinary derivative of the single-variable function $g(x) = f(x, b)$.

Geometrically, $\partial f / \partial x$ at $(a,b)$ is the slope of the tangent line to the curve obtained by slicing the surface $z = f(x,y)$ with the plane $y = b$. Each partial derivative gives information in one coordinate direction only.