Overview
The partial derivative is the normal derivative of a single variable function applied to the multi-variable function by holding all variables constant except for the variable for which you are taking the partial derivative.
{% \frac{\partial{f(x_1,...,x_n)}}{\partial{x_i}} %}
Definition
For a multi-variable function
{% f(x_1,...x_n) \rightarrow \mathbb{R} %}
the partial derivative with respect to the variable {% x_i %} is given as
{% \frac{\partial{f(x_1,...,x_n)}}{\partial{x_i}} = \lim_{s \to 0} \frac{f(x_1, ...,x_i + s,...x_n) - f(x_1, ...,x_i,...x_n)}{s} %}