Q: OpenMDAO: When is it needed to define the partial derivative? I've noticed that defining unneccesary partial derivatives can significantly slow down the optimizer. Therefore I'm trying to understand: how can I know whether I should define the partial derivative for a certain input/output relationship?

A: I think they need to be defined if they are ever relevant to a response (constraint or objective) in the optimization, or as part of a nonlinear solve within a group.  My personal practice is to always define them.  Should I every change my optimization problem, which I do often, I don't want to have to go back and make sure I'm always defining the appropriate derivatives.
The master-branch of OpenMDAO contains some jacobian-coloring techniques which can significantly improve performance if your problem is particularly sparse in nature.  This method is enabled by setting the following options on the driver:
p.driver.options['dynamic_simul_derivs'] = True
p.driver.options['dynamic_simul_derivs_repeats'] = 5

This method works by filling in the user-described sparsity pattern (specified with rows and cols in declare partials) with random numbers and computing the total jacobian.  The repeat option is there in improve confidence in the results, since it's possible but unlikely that a single pass will result in an "incidental zero" in the jacobian that is not truly part of the sparsity structure.
With this technique, and by doing things like vectorizing by calculations instead of using nested for loops, I've been able to get very good performance in a lot of situations.  Of course, the effectiveness of these methods is going to change from model to model.

A: When you say "unnecessary" do you mean partial derivatives that are always zero? 
Using declare_partials('*', '*'), when a component is really more sparse than that will significantly slow down your model. Anywhere where a partial derivatives is always zero, you should simply not declare it. 
Furthermore, if you have a vectorized operation, then your Jacobian is actually a diagonal matrix. In that case, you should declare a [sparse partial derivative] by giving rows and cols arguments to the declare_partial call1. This will often substantially speed up your code. 
Technically speaking, if you follows the data path from all of your design variables, through each components, to the objective and constraints, then any variable you passed would need to have its partials defined. But practically speaking you should declare and specify all the partials for every output w.r.t. every input (unless they are zero), so that changes to model connectivity don't break your derivatives. 
It takes a little bit more time to declare your partials more sparsely, but the performance speed up is well worth it.  
