partial derivative

a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

let $\text{[math]}$ be an open subset of $\text{[math]}$ and $\text{[math]}$ a function. the partial derivative of $\text{[math]}$ with respect to the $\text{[math]}$ th variable, and evaluated at $\text{[math]}$ , is the limit
$\text{[math]}$ if the limit exists.
[cite:@calc_hubbard_2015 definition 1.7.3]

the partial derivative for an $\text{[math]}$ -valued (or "vector-valued") function, a function $\text{[math]}$ from $\text{[math]}$ to $\text{[math]}$ , is
$\text{[math]}$ which is an $\text{[math]}$ matrix.
[cite:@calc_hubbard_2015 equation 1.7.6]