First teach what the network does and why, writing it as a loopy, inference-only Python function. Explain training only in an abstract way, E.G. with the "take a random weight, twist it a little and see if the loss improves" algorithm. This lets you focus on the architecture and on why it is what it is.
Then, teach the intuitions behind derivatives and gradient descent. You don't need the entirety of calculus, there's no benefit to knowing how a sequence or limit works if you ) only want to understand neural networks. With autograd, you won't be manually doing derivatives of weird functions either, so intuitive understanding is a lot more important than doing dozens of traditional calculus exercises on paper like it's the 1800s. You could probably explain the little bit of calculus you need in an hour or two, even to somebody with a 12-year-old's understanding of math and a good bit of programming knowledge.
Only when people understand the training and inference, implemented with loops and descriptive variable names, teach the tensor, explain how a modern CPU and GPU works (because many programmers still think a modern computer is just a much faster 6502), and then teach the tricks we use to make it fast.