The Divinity of Jesus
It is crucial to first understand traditional Christology, and so the meaning of the doctrine 'Jesus is God.'
According to traditional Christianity, that Jesus is God means He is, as to His nature, θεος (the Greek word for 'God'). St. John opens His Gospel thus:
John 1:1 (DRB)
In the beginning was the Word,
and the Word was with God,
and the Word was God.
From the context you can see this doesn't mean 'the Word was with the Father and was the Father,' but that the Father, also known as God, is God, as is the Son.
This could only be possible if He shares the same nature or essence or substance (or 'what makes it what it is') of God the Father, “the only true God,”1 and as such is not a different God from Him, but the same God: in the words of the Nicene Creed: “God of God, Light of Light, very God of very God; begotten, not made; of one substance with the Father.”
The Hypostatic Union
The hypostatic union is the doctrine that the “Word [who is] God”1 has proper to Him two natures: the divine nature and the human nature are united in only one Person,2 the divine Word. (And importantly, not a hybrid nature ('neither truly fully man nor truly fully God'),3 but fully possessing each as a complete and independant nature in and of itself.)
If this sounds complicated, it is simply the teaching that “the Word [who is God] became flesh and tabernacled among us.”4 And so in “taking”5 human flesh, He did not change into another person, nor is another person introduced in the Incarnation event (i.e. no 'divine Son' and 'human Jesus'), but the one divine Word assumes a nature He did not before have for the purposes of Redemption as the New Adam 'from heaven, not from the dust:'6 “In the fulness of time, God sent forth His Son, born of woman...”7 The same Son, or Word, of God that had it in “mind”8 to become Incarnate is the same Son who was incarnated. He proceeded His birth, “from ancient times, from the days of eternity.”9
The issue at hand is who died at Calvary, not what died: did the Word take a human nature in which He cannot die, or one in which He can and did?
1 Corinthians 2:8 (DRB)
[If the princes of this world had known the hidden wisdom of God] they would never have crucified the Lord of glory.10
Whatever is proper to the human nature Jesus took, or His divine nature, is proper to one and the same Jesus regardless.
Thus, whereas God cannot die in His divine nature, He, a divine person taking on a human nature, can die in a human nature, in which it is possible for Him to experience death.
Revelation 1:17-18 (DRB)
And when I had seen him, I fell at his feet as dead. And he laid his right hand upon me, saying: Fear not. I am the First and the Last, 18 And alive, and was dead, and behold I am living for ever and ever, and have the keys of death and of hell.
Thus, the First and the Last can be born (and thus have a mother) and die precisely and only because He took on a real human nature. So the negative sentiment underlying 'your God died' arguments fail to see that dying is an action God can do when He takes on a human nature—it had nothing to do with the quality of the divine nature or His abilities (i.e. the implied 'your God can be hurt' behind the argument). It's ineffably beautiful rather, that God, for whom it is laughably impossible to hurt or feel pain, went out of His way to be able to feel pain and be tortured, just to save a plethora of ungrateful wretches who spat at Him physically and spiritually at every moment while He was in such a state—just as He knew when He underwent it to redeem us from the fate we merited freely by our sin.
On one hand, it's a great miracle that the Word became flesh. On the other hand, it's not so unimaginable or impossible that the Creator of people (i.e. persons) could, instead of creating a soul with a new person, create a soul and body for a[n already existent] Person (the Word, His Son).11
It's tremendously easy for God to become flesh: what is amazing is that He did and what such a decision implies about His love for us.
1 John 17:3.
2 'Hypostatic union' contextually means 'the unity of person.' That is, the divine nature of Christ and His assumption of a human nature does not in any way imply a second 'person.' There is only one person, not two 'Jesuses.'
3 A heresy known as Monophysiticism, which is a Greek-derived term meaning 'One-natureism.' That is, in the Incarnation, the the divine and human nature were 'fused,' as it were,' into one new nature: 'divine-human nature.'
4 John 1:1, 14.
5 Philippians 2:7.
6 1 Corinthians 15:47.
7 Galatians 4:4.
8 Galatians 4:4.
9 Micah 5:2; cf. Matthew 2:6.
10 Psalm 24:8.
11 cf. Hebrews 10:5.