To give a common historical-critical perspective (one that I don't necessarily share in all details; but again, a common one): the earliest Christians would not have associated Jesus' death on the cross with his "full divinity," as it was fleshed out in the later Christologies of the 2nd-4th century.

The problematic aspects of reading these later Christologies back into the Biblical texts can be illustrated by several Pauline examples. The first is the famous Philippians "hymn" (2:5-11):

5 Let the same mind be in you that was in Christ Jesus, 6 who, though existing/living in the form of God, did not regard equality with God as something to be exploited, 7 but emptied himself, taking the form of a slave, being born in human likeness. And being found in human form, 8 he humbled himself and became obedient to the point of death-- even death on a cross. 9 Therefore God also highly exalted him and gave him the name that is above every name, 10 so that at the name of Jesus every knee should bend, in heaven and on earth and under the earth, 11 and every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.

If Jesus really had "equality with God," then how could God have "highly exalted him and gave him the name that is above every name"? This was recognized to be problematic in the early church. For example, the important-fourth century Church Father and Trinitarian Athanasius argues that

[i]t was the Word's humanity that was exalted . . . since the Word is always divine and needs no exaltation. Statements such as these, Athanasius explains, are made "humanly" (ἀνθρωπίνως), with reference to the flesh that the Word took on, whole others are said "divinely" (θεϊκῶς), such as "the Word was God" . . . Athanasius presses the distinction so far as to say that the human statements do not really apply to the Word but to us, and Philippians 2 does not indicate that the Word is exalted, but that we are exalted (C.Ar. 1.41)

It's clear that the interpretation that things like Philippians 2:9 are only saying that "we are exalted" is totally untenable.

Yet can we really even say that it was only Jesus' humanity that was exalted? Here, we'd also come up against other problematic verses -- like Romans 1:4, where Jesus was "designated (or even became) Son of God . . . by resurrection from the dead" (you can see this thread for more on Romans 1:4). Yet does this suggest that Christ was not always the Son? Although even modern academic exegetes of this passage can suggest more of a focus here on a "functional" rather than an "ontological" "adoptionism" (which can certainly be reconciled with later orthodox Christology), I think we're more warranted in the interpretation that Jesus really did take on "a status and role that he did not have previously" here.

The ancient evidence is absolutely unequivocal that "human" and "divine" resided on a spectrum; and the evidence suggests that the earliest Christianity was no different (in regards to how Christ's divinity was construed) -- and thus anachronistically superimposing later Christologies back onto the earliest strata of Christianity would be an error.