If a function has more than one variable, say , we are able to take a partial derivative of the function in respect to one of its variable:
Find the first order partial derivatives for the function:
First take the derivate with respect to x treating all coefficients with a y as a constant. The partial derivative with respect to x is:
Take note that the second and third terms in the equation differentiate to zero when the partial derivate with respectto x is taken, so they are eliminated.
The next step is to take the partial derivative with respect to y while treating all the coefficients with an x as a constant. The partial derivative with respect to y is: