Derivatives & Differential Equations
The ordinary derivative
of a function of one variable can be carried out because everything else in the
function is a constant and does not affect the process of differentiation. When
there is more than one variable in a function it is often useful to examine the
variation of the function with respect to one of the variables with all the
other variables constrained to stay constant. This is the purpose of a partial
derivative. The derivative is carried out in the same way as ordinary
differentiation with this constraint. For example, given the polynomial
in variables x and y,
the
partial derivative with respect to x is written
and the partial derivative with respect
to y is written