(I am using the word "fetus" to denote the whole stage from fertilisation to birth. I am using the American spelling because the English only added the "o" to be pretentious, and the Americans then rightly removed it again.)
There are several important questions which ought to be asked about the fetus. Is it alive? Is it human? Is it individuated (i.e. a distinct human from its mother)?
"Alive" seems to be a binary state. Things are either alive or they are not. We do not see examples of things in everyday life which are "partially alive". We sometimes speak like that when things are in danger of dying, but that is not the issue here. The question is not alive/dead but living/non-living. There are also examples in nature of systems which are partially living, such as soil, but that is because they are composed of a large number of small living things, with other bits of dead and non-living material. That description does not apply to a fetus.
So is the fetus living? It is certainly living at the moment of birth. If it was non-living at any moment before then, there must have been a sudden point where it made the transition from non-living to living. We do not observe such a point in pregnancy. Therefore the fetus is living from the time of fertilisation to the time of birth.
Is the fetus human? This, of course, depends on the definition of "human". Our definition needs to include all instances of humanity, including babies and people who are asleep or in comas, which means that the definition would either have to be in terms of potential, e.g. "A human is a physical being which, at some stage in the past or future, might be able to tell a joke / relate to God / whatever." or in terms of genetics. All of these definitions apply to fetuses throughout pregnancy, unless fetuses are specifically and directly excluded from them. Hence it seems logical to conclude that fetuses are human.
Are fetuses individuated or are they "just part of the mother"? The same argument seems to hold as for living. Things are not partially individuated. Indeed, the use of the word "mother" above tells you that there are two distinct individuals - the mother and the fetus. If they were one individual, she would not be a "mother", and there would be no "father" either. And so, because they are individuated at birth, and there is no one moment of individuation, they must also be individuated at conception.
This does not of course mean that they are physically distinct. Indeed, the situation is in many ways comparable to the situation of conjoined twins, where there are two individuals in one composite body. Except in this case, of course, the body is mostly that of the mother.
So then, it seems most logical to me to conclude that the fetus is an individuated, but not physically distinct, living human.