Generally, the side presenting the expert will have spent considerable time laying a foundation for the expert witness's expertise: their credentials, education, publications, and experience in the field about which they are testifying.
The opposing side may try to attack the expert's credentials. Maybe they have book knowledge, but little real-world experience. Perhaps their studies have been on a related subject but not directly about the matter in dispute at trial. Regardless, to the extent possible, the opposing attorney will try to minimize the expert's impact by challenging his or her expertise.
Before doing that, however, the opposing counsel should try to use the expert to advance his or her own case. For example, if a defense expert can offer anything in support of the prosecution's case, the prosecutor should try to get that information before the jury. It is very powerful evidence for a witness engaged and called by the other side to give favorable testimony for your case.
Eliciting any favorable testimony should be done before attacking the expert's testimony because the impact of the expert's testimony will be lessened after he or she has been discredited (hopefully).