Partial Derivative

Definition

Let $f:\mathcal{D}\subset {\mathbb{R}}^n\to \mathbb{R}$ be a differentiable, and $\mathbf{p}\in \mathcal{D}$. Then, the partial derivative of $f$ at the point $\mathbf{p}$ with respect to $i$-th variable is defined as $\frac{\partial }{\partial x_i}f(\mathbf{p}):={\nabla }_{{\mathbf{e}}_i}f(\mathbf{p})={\lim }_{h\to 0}\frac{f(\mathbf{p}+h{\mathbf{e}}_i)-f(\mathbf{p})}{h}$ where ${\mathbf{e}}_i$ be a Standard Basis