# Direct Differentiation

In this approach, the equations of motion are analytically differentiated with
respect to each design variable *b*_{i}, i=1…* N*_{b}.

At each point in time, after an analysis step is complete, *N*_{b} sets
of DAE are solved numerically to yield the required values
$\frac{\Delta y}{\Delta b}$
. This approach is quite robust and is used when the
numbers of design variables *N*_{b} are not too large. As the number of
design variables increase, the cost of the solution also increases.