The chain rule with one picture

TL;DR:

• For y = f(g(x)), the derivative is f′(g(x)) · g′(x): outer derivative times inner derivative.

• The reason is that rates multiply along a chain of dependencies.

• Check with a small power you can expand by hand.

The power rule handles x⁵. But what about (3x + 1)⁵? You could expand it into six terms, but for (3x + 1)⁵⁰ that is hopeless. The chain rule gives the answer in one line, and the idea behind it is something you already know from everyday ratios.

A chain of two machines

Think of two machines in a row. The inner machine takes x and produces u = 3x + 1. The outer machine takes u and produces y = u⁵. When x changes a little, u changes, and that makes y change.

If a small change in x makes u change at a rate of du/dx, and a small change in u makes y change at a rate of dy/du, then the overall rate of y with respect to x is the product: dy/dx = dy/du × du/dx. It works like gears: if the first gear turns the second 3 times as fast, and the second turns the third 5 times as fast, then the first turns the third 15 times as fast.

Worked example: d/dx (3x + 1)⁵

1. Name the inner function: u = 3x + 1. Its derivative is du/dx = 3.

2. Name the outer function: y = u⁵. Its derivative is dy/du = 5u⁴.

3. Multiply: dy/dx = 5u⁴ × 3 = 15u⁴.

4. Put the inner function back: dy/dx = 15(3x + 1)⁴.

Check at x = 0. The rule gives 15 × 1⁴ = 15. A numeric test: at x = 0.001, (1.003)⁵ ≈ 1.015090, so the slope is about 0.015090 / 0.001 = 15.09. It heads toward 15.

Check by expanding a smaller power

To trust the method, test it where you can expand by hand. Take (3x + 1)². Expanding gives 9x² + 6x + 1, whose derivative is 18x + 6. The chain rule says: inner derivative 3, outer derivative 2u = 2(3x + 1), product 6(3x + 1) = 18x + 6. They match.

Another pair: (x² + 1)³

Inner: u = x² + 1, with du/dx = 2x. Outer: y = u³, with dy/du = 3u². The product is 3(x² + 1)² × 2x = 6x(x² + 1)². At x = 1, that is 6 × 1 × 4 = 24. Check by expanding: (x² + 1)³ = x⁶ + 3x⁴ + 3x² + 1, whose derivative is 6x⁵ + 12x³ + 6x, and at x = 1 it is 6 + 12 + 6 = 24. Same.

The rule beyond powers

The outer function does not have to be a power. For sin(x²), the inner function is x² with derivative 2x, and the outer is sine, whose derivative is cosine (when angles are in radians). So the derivative is cos(x²) × 2x. The recipe is always the same: outer derivative evaluated at the inner function, times the inner derivative.

Try one yourself: (2x² − 3)⁴

The inner function is u = 2x² − 3, with du/dx = 4x. The outer function is u⁴, with derivative 4u³. Multiply: 4(2x² − 3)³ × 4x = 16x(2x² − 3)³. At x = 2, u = 5, and the derivative is 16 × 2 × 125 = 4000.

Check with numbers. At x = 2, the function is 5⁴ = 625. At x = 2.001, u = 5.008002 and u⁴ ≈ 629.01. The slope over that small step is about (629.01 − 625) / 0.001 ≈ 4010. That is a little above 4000 because the slope is still increasing across the step, and it closes in on 4000 as the step shrinks.

If y is in metres, u in seconds and x in litres, then dy/du is metres per second and du/dx is seconds per litre. Multiplying gives metres per litre, which is what dy/dx should be. The middle unit cancels, just as in a unit conversion. This is a strong reason to trust that the rates multiply.

Reading it as a rate problem

Suppose the temperature in a room depends on the heater setting, and your comfort score depends on the temperature. If 1 notch on the heater adds 2 degrees, and each degree adds 3 points of comfort, then 1 notch adds 2 × 3 = 6 points. These are invented numbers, but they show why the rates multiply rather than add.

How to check your answer

• Test a small power. Replace the exponent by 2 and compare with an expansion.

• Numeric slope. Evaluate the function at two close points.

• Count the pieces. The answer must contain both an outer part, (3x + 1)⁴, and an inner part, 3.

• Degree check. The exponent drops by one from the original.

Common mistakes

• Forgetting the inner derivative, writing 5(3x + 1)⁴ instead of 15(3x + 1)⁴.

• Differentiating the inner function inside the outer one, giving 5(3)⁴.

• Applying the chain rule when it is not needed, such as to a plain x⁵.

• Mixing up the order of the names and writing du/dx where dy/du belongs.

Where you will use it next

The chain rule is the engine behind related rates: the sliding ladder, where every quantity depends on time. It builds on the power rule. Any time one changing thing depends on another, expect a chain.

Frequently asked questions

What if there are three functions nested, like sin((3x + 1)⁵)?

Multiply three derivatives: outer cosine, middle power, inner line. Work from the outside in, one layer at a time.

Is the chain rule the same as the product rule?

No. The product rule is for two functions multiplied together, like x·sin x. The chain rule is for one function inside another.