Let $f : \mathbb{R}^n \to \mathbb{R}$ and $\mathbf{a} \in \mathbb{R}^n$. The partial derivative of $f$ with respect to $x_i$ at $\mathbf{a}$ is $\frac{\partial f}{\partial x_i}(\mathbf{a}) = \lim_{h \to 0} \frac{f(\mathbf{a} + h\mathbf{e}_i) - f(\mathbf{a})}{h}$ where $\mathbf{e}_i$ is the $i$-th standard basis vector. This measures the rate of change of $f$ in the direction of the $x_i$-axis while holding all other variables fixed.

The gradient of $f$ at $\mathbf{a}$ is the vector of all partial derivatives: $\nabla f(\mathbf{a}) = \left(\frac{\partial f}{\partial x_1}(\mathbf{a}), \ldots, \frac{\partial f}{\partial x_n}(\mathbf{a})\right)$ The gradient points in the direction of steepest ascent and its magnitude gives the maximum rate of change: $|\nabla f(\mathbf{a})| = \max_{\|\mathbf{u}\|=1} D_\mathbf{u} f(\mathbf{a})$.

A function $f : \mathbb{R}^n \to \mathbb{R}$ is differentiable at $\mathbf{a}$ if there exists a linear map $L : \mathbb{R}^n \to \mathbb{R}$ such that $\lim_{\mathbf{h} \to \mathbf{0}} \frac{f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) - L(\mathbf{h})}{\|\mathbf{h}\|} = 0$ The linear map $L$ is the total derivative (or differential) $Df(\mathbf{a})$, and $L(\mathbf{h}) = \nabla f(\mathbf{a}) \cdot \mathbf{h}$.