Let $$ f$$ be a function and let $$ a$$ be a point in its domain, then the derivative of f at a is defined as: $$ \underset{h\to <!--\; \to \; -->0}{lim}\frac{f(a+h)-<!--\; -\; -->f\left(a\right)}{h}$$, 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 $$ {\mathbb{R}}^n$$, and let $$ f:U\to <!--\; \to \; -->\mathbb{R}$$ be a function, then the partial derivative with respect to the $$ i$$-th variable, $$ x_i$$, of $$ f$$ at $$ a=(a_1,\dots <!--\; \dots \; -->,a_n)\in <!--\; \in \; -->U$$ is defined as: $$ \underset{h\to <!--\; \to \; -->0}{lim}\frac{f(a_1,\dots <!--\; \dots \; -->,a_i+h,\dots <!--\; \dots \; -->,a_n)-<!--\; -\; -->f(a_1,\dots <!--\; \dots \; -->,a_i,\dots <!--\; \dots \; -->,a_n)}{h}$$ when it exists, and is denoted by $$ \frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}{x}_i}f\left(a\right)$$