Owl: A livestream's "buzz score" is the product of two changing counts: active viewers V(t) and average emoji reactions per viewer E(t). Both are rising. How fast is the product B = V·E changing?
Your gut might say "multiply the two rates." That gut is wrong — and the product rule exists because of it.
Differentiation rules are accelerators: power, constant multiple, sum, product, quotient. They don't replace the meaning of derivative; they compute it faster once you trust the limit definition.
We'll verify a rule numerically on a tiny example, then ask why (uv)' is not u'v'. What's your first guess for how the product should grow when both factors move?