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Let $f$ be a function and let $a$ be a point in its domain, then the derivative of f at a is defined as: ${\lim }_{h\to 0}\frac{f(a+h)-f\left(a\right)}{h}$, when it exists and is denoted by $f^{{}^{\prime }}\left(a\right)$ or $\frac{d}{dx}f\left(a\right)$

Let $U$ be an open subset of ${ℝ}^n$, and let $f:U\to ℝ$ be a function, then the partial derivative with respect to the $i$-th variable, $x_i$, of $f$ at $a=(a_1,\dots ,a_n)\in U$ is defined as: ${\lim }_{h\to 0}\frac{f(a_1,\dots ,a_i+h,\dots ,a_n)-f(a_1,\dots ,a_i,\dots ,a_n)}{h}$ when it exists, and is denoted by $\frac{\partial }{\partial x_i}f\left(a\right)$