Before the method: the idea behind it
Every derivative you've found so far has belonged to a single, self-contained function. Almost nothing in the real world works that way. Quantities are usually built by multiplying other changing things together, or by hiding one changing thing inside another. This page is about why that matters, before we touch a single algorithm.
Part one
Picture a community garden bed getting bigger at the start of spring. Each week the gardeners make it a little wider and a little longer.
If only the width were changing, this would be easy: multiply the width's rate of change by the fixed length and you're done. But here, both sides move at once. Growing the width adds new area along one edge. Growing the length adds new area along the other edge. And right in the corner, where both changes overlap, a tiny new patch appears too.
That's the whole idea of the product rule, before any algebra. When two things that are multiplied together both change, the total rate of change is made up of two separate contributions: one from each factor doing its own changing, while the other factor is momentarily held still.
A garden bed of width \(w\) and length \(h\), about to grow on both sides.
Here's what to watch for. As the width grows by a small amount and the length grows by a small amount, three new patches of area appear: a strip along the right edge, a strip along the top edge, and a tiny square in the corner where the two strips overlap. As the growth gets smaller and smaller, that corner square shrinks away much faster than the two strips do. In the limit, only the two strips are left. That's the exact shape of the product rule.
In plain terms: the total rate of change is the width's own rate of change scaled by the current length, plus the length's own rate of change scaled by the current width. Drop either term and the picture is incomplete. Both contributions are needed.
A shop's revenue is price times quantity sold. If it raises its price over time, and the number of items it sells is also changing, the revenue's rate of change combines both effects: \(R'(t) = P'(t)Q(t) + P(t)Q'(t)\). This is why raising the price doesn't always raise revenue. If sales are falling off fast enough, the second term can outweigh the first.
Whenever a total is built from two things multiplied together and both are moving, the reach of a marketing campaign (people reached times response rate), or the output of a farm (area planted times yield per hectare), the product rule tells you how the whole system's rate of change splits between the two causes.
Part two
Now picture a weather balloon lifting off the ground, a thermometer strapped to its basket.
From the ground, all you can see is the temperature falling as the seconds tick by. But two separate relationships are hiding behind that single number. The balloon's height depends on time, and the temperature depends on height. Time never touches temperature directly. It only gets there by changing the height first.
This is what the chain rule captures: when one function is nested inside another, its overall rate of change is the product of the rates along each link of the chain.
Height depends on time. Temperature depends on height. The thermometer only ever shows you the end result.
Each derivative on the right only knows about one link: how temperature responds to height, and how height responds to time. Multiplying them together lets that information travel all the way from time to temperature, through the hidden middle step. It's also why it's called the chain rule. Every link has to be accounted for, and the links connect by multiplying, not adding.
The radius of a spreading circular spill depends on time, and the area of the spill depends on the radius. Environmental teams use this exact chain, radius over time and area over radius, to work out how fast the contaminated area is growing, without ever writing area directly in terms of time.
Altitude affects air pressure. Depth affects water pressure. Exchange rates affect the local cost of an imported good. Anywhere one changing quantity is defined in terms of another changing quantity, the chain rule is how their rates of change connect.
Before you start
Both rules exist because a function is built from more than one moving part. The difference is how the parts fit together.
| Ask yourself… | If yes, this is… |
|---|---|
| Are two different functions of \(x\) multiplied together, side by side? | Product rule |
| Is one function sitting inside another, like a wrapper: a bracket raised to a power, or a root of an expression? | Chain rule |
| Could it be both at once, e.g. \(x^2(3x-1)^4\)? | Both together, coming later |
Next, we'll turn the product rule's idea into a step by step method you can apply to any question, starting with the simplest cases and building up.