# Direct Differentiation

In this approach, the equations of motion are analytically differentiated with respect to each design variable bi, i=1… Nb.

At each point in time, after an analysis step is complete, Nb 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 Nb are not too large. As the number of design variables increase, the cost of the solution also increases.