Ask most students what the chain rule is, and you'll get "derivative of the outside times the derivative of the inside." Technically correct. Almost useless as an intuition, because it doesn't explain why that's true — which means the moment the "outside" and "inside" aren't obvious, the rule stops helping.
Think in rates, not rules
Here's the picture: a derivative is a rate of change — how fast one quantity changes relative to another. The chain rule is just what happens when you stack two rates of change on top of each other.
Suppose your paycheck depends on hours worked, and hours worked depends on how many days you're in town. If your pay changes at a rate of $20 per hour, and your hours change at a rate of 8 hours per day, then your pay changes at a rate of 20 × 8 = $160 per day. You didn't need a rule to know that — you just multiplied two rates that were chained together.
The chain rule isn't a trick for derivatives. It's just what "rates of change" do when you stack them.
Mapping it to the formula
If y depends on u, and u depends on x, then:
dy/dx = (dy/du) × (du/dx)
Read left to right, that's exactly the paycheck example: the rate y changes with respect to x is the rate y changes with respect to u, times the rate u changes with respect to x. The units even cancel the way you'd expect — the "du" in the numerator of one fraction cancels the "du" in the denominator of the other.
Why this matters more than the mechanical rule
Once you see the chain rule as "multiplying stacked rates," it stops mattering how deeply nested the function is. Three functions chained together? Multiply three rates. A function inside a trig function inside an exponential? Same idea, just more links in the chain.
Students who memorize "outside times inside" often get stuck the first time they see something like sin(ln(x²)) because it's not obvious what's "outside" and what's "inside." Students who understand it as stacked rates just peel off one layer at a time, multiplying as they go.
Try it yourself
Next time you hit a chain rule problem, pause before differentiating and ask: what are the rates being stacked here, and what does each one represent? If you can answer that in plain language, the derivative is just bookkeeping.