The response to this question lies in understanding that the outcome of anything is dependent on the performance of the efforts that are necessary to procure it. In other words, it has already been decreed, for example, that a seed will give fruit if planted, but this will not occur unless the farmer takes the appropriate efforts in irrigating the crop, maintaining it, protecting the seedling as it grows, and ensuring as much as he can that the factors are amenable for the plant to give fruit.
So even though a person believes in the Divine Decree, he must at the same time strive all he can to ensure that the desired goal occurs. So du'aa' is the means that one uses to achieve the desired goal that one has, and this means in no way contradicts the destiny that has been written for that person.
This stance is clarified by the Prophet's صلى الله عليه وسلم hadeeth reported by Thawban, that he صلى الله عليه وسلم said: "Nothing increases one's lifespan except good deeds, and nothing repels Divine Decree except du'aa'. And verily, a person may be deprived of sustenance due to a sin that he does!" 
In other words, the performance of good deeds is a cause of increasing one's life span, so if a person puts in the necessary effort, the results will be achieved, and this is also destined. So both the means to achieve a goal, and the fulfillment of the goal itself, are already decreed.
If someone were to ask: "How can du'aa' repel Divine Decree?" we would respond, "The fact that you may be ill has already been decreed by Allaah سبحانه و تعالى for you, as has the fact that you will ask Allaah سبحانه و تعالى to cure you of this illness (i.e. both the illness and your asking to cure the illness has been decreed). In a similar manner, a person may be deprived of his sustenance that was decreed for him, since Allaah's eternal knowledge encompassed the fact that this person would perform a sinful deed that would cause the deprivation of his sustenance. All of this, then, is from the decree of Allaah, all Glory and Praise be to Him".
There are a number of ahaadeeth that clarify this point. For example, Mu'aadh ibn Jabal رضي الله عنه reported that the Prophet صلى الله عليه وسلم said: "Caution will be of no benefit against Divine Decree, but du'aa' benefits all things, whether they come down or not. I therefore advise you to make du'aa', O servants of Allah!" 
So no matter how cautious a person is, he cannot escape what is written for him, simply because Allaah سبحانه و تعالى controls everything, and nothing escapes His Knowledge or Power. However, by turning to Allaah سبحانه و تعالى through du'aa' it is possible to avert something that might have been decreed. Salmaan al-Faarisi رضي الله عنه narrated that the Prophet صلى الله عليه وسلم said: "Nothing repels Divine Decree except du'aa', and nothing increases one's life-span except good deeds." 
This hadeeth informs us in no uncertain terms that the only way that we can repel some Divine Decree is through the means of du'aa'. So it is possible that some unpleasant matter has been preordained for us, but only if we do not make du'aa' to avert it from us. So if du'aa' is made, then this matter will not occur or be fulfilled, whereas if du'aa' is left, the misfortune will occur.
Another narration in at-Tirmidhi also supports this. The Prophet صلى الله عليه وسلم said: "There is no Muslim on the face of the earth that asks Allaah for anything except that Allaah gives it to him, or averts from him a similar evil, as long as he does not ask for something evil or for breaking the ties of kinship." 
From this narration, the benefits of du'aa' are made clear, and its value is understood. For not only is a person rewarded for making a du'aa' but it is also a cause of repelling an evil that was destined for him, and in obtaining the good that he was expecting.
Imaam Ibn Hajr رحمه الله, commenting on the benefits of du'aa', said: "And the benefit of performing du'aa' is the attainment of reward by obeying the command (of Allaah سبحانه و تعالى to make du'aa'), and also by the attainment of what is asked for, for there is a possibility that the request is dependent on the du'aa', since Allaah سبحانه و تعالى is the Creator of both the effort and result of the effort!" 
Therefore, the proper response to the question posed at the beginning of the chapter is, in the words of Imaam Ibn al-Qayyim رحمه الله, as follows: "The logical consequences of such a reasoning leads to a rejection of all efforts. It can, therefore, be said to a person who holds this view, 'If the satisfaction of your hunger and thirst has already been destined for you, then it will be fulfilled, whether you eat or not. On the other hand, if it has not been destined for you, it will never occur, whether you eat or not. And, if a son has been destined for you, then you will be granted one, whether you have intercourse with your wife or not. On the other hand, if a son has not been destined for you, then you will never be granted one. In this case, there is no point in you getting married!' "Now, will any sane person agree with all of these conclusions?" 
To summarise, then, Divine Decree (qadr) cannot be used as an excuse not to make du'aa'. For, just as one strives to ensure that one attains worldly needs, of food, drink and family, so too must one strive in one's religious deeds to attain the desired goal. du'aa' is intrinsically related to qadr; in fact, it is a part of one's qadr. Allaah سبحانه و تعالى has already decreed that a certain matter will be granted to a servant, or an evil averted from him, if he makes du'aa', so if he were to leave du'aa', then the desired goal would not be reached.
Hence why it was the Sunnah of the Prophet صلى الله عليه وسلم to make the following du'aa' during the witr prayer: "...Bless me in what You have given me. And avert and turn away from me the evil that has been decreed for me, for verily You Decree (all things), and none can decree against You. . ." 
So the Muslim turns to Allaah سبحانه و تعالى and prays to Him so that any and all evil can be averted from him.
Additionally, it should be remembered that the concept of Divine Decree is one that a human can never fully understand, due to his limited intellect and finite capabilities. The true Muslim does not delve too deeply into the philosophical ramifications of Divine Decree. Rather, he accepts all that has occurred to him in the past as having been destined for him, and he strives to obtain what he desires in the future (as long as it is permissible for him). Since he does not know what has been written for him in the future, he expects the best from Allaah سبحانه و تعالى, and does everything in his power to ensure that what he wants is granted him. Just like going to work every day will ensure, if Allaah wills, that he gets his pay-cheque at the end of the month, so too does making du'aa' ensure, if Allaah wills, that he achieves his desired goal. 
 Narrated by Ibn Maajah #90, and Shaykh al-Albaani رحمه الله said in Saheeh Ibn Maajah (73): "It is authentic without the addition, 'And verily...'; see as-Saheehah, #154.
 This hadeeth is weak. It was narrated by Ahhmad, Abu Yala, and at-Tabaraani in al-Kabeer, as has been mentioned in Da'eef al-Jaami' #4785.
 Authentic, reported by at-Tirmidhi and al-Haakim from Sayyiduna Salmaan رضي الله عنه, and is in Saheehul Jaami' #7687.
 Authentic, narrated by at-Tirmidhi from Sayyiduna 'Ubaadah Ibn as-Saamit رضي الله عنه, as is mentioned in Saheehul Jaami' #5637.
 Fat-hul Baari, 11/95
 al-Jawaab al-Kaafi, p.13
 Reported by at-Tirmidhi (#464), an-Nasaa'ee (#1725) and others, with an authentic chain.
 Dua: The Weapon of the Believer.