Partial Derivative
Statements
Definition. (@Boo75 pp. 21) Let be an open set and consider the function
f:&U&\to&\mathbb{R}\\ &x&\mapsto&f(x)\;. \end{array}$$For every $a\in U$, the _partial derivative of $f$ at $a$ with respect to $x^j$_ is given by the limit $$\left.\dfrac{\partial f}{\partial x_j}\right|_a=\displaystyle\lim_{h\to 0}\dfrac{f(a^1,\ldots,a^j+h,\ldots,a^n)-f(a^1,\ldots,a^j,\ldots,a^n)}{h}$$ if it exists.