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The Chain Rule

Every rule in the previous lesson handles functions built by adding, multiplying or dividing. The chain rule handles the one remaining way to build a function: putting one inside another. That covers most of what you will actually meet, so this rule gets used more than all the others together. The rule itself is a single line; the skill is seeing that you are looking at a composite in the first place.

Calculus — differentiation

Graphing calculator
The rule
  • Chain rule

    Differentiate the outside, leave the inside exactly as it was, then multiply by the derivative of the inside. That last factor is the part people drop.

  • In Leibniz notation

    With the inside. Written this way it looks like cancelling , which is a useful memory hook and not a proof.

  • Why there is a second factor

    Expand and check: , whose derivative is . The inside runs three times as fast as , so the whole thing does too. Dropping ignores that.

  • The power case

    The most common form by far. Power rule on the shell, then times the derivative of what is inside it.

  • With the standard functions

    And , which is the same rule with as the outer derivative.

When the inside is just , its derivative is 1 and the extra factor changes nothing — which is why the rules in the last lesson looked like they had no chain factor. They did; it was invisible.

Finding the inside and the outside
Getting the layers the right way round is the whole job. One question settles it every time.

  1. Ask what you would do last. If you had to work out for on a calculator, you would square first and take the sine last. The thing you do last is the outside; everything you did before it is the inside.

  2. Look for a shell. A bracket raised to a power, or , , , , with something more than a bare in it. That is a composite and it needs the rule.

  3. Watch the notation that hides it. means — a cube on the outside, cosine inside. It is a composite even though nothing is visibly nested, and it is a different function from .

  4. Peel one layer at a time. There can be three. is sine of a root of a polynomial, and each layer contributes a factor. Take the outermost, then treat the rest as a new problem.

  5. Check the answer contains g'. The derivative of the inside must appear as a factor somewhere. If your answer has no trace of it, you have dropped it — that is the single most common error in this topic.

  6. Do not differentiate the inside in place. The inside stays exactly as it was inside the outer function. starts , never .

Worked examples — one layer
Each one names the outside and the inside before any differentiating, which is the habit worth copying.

Worked examples — with the other rules
Real questions stack the rules. Decide the outermost structure first — is the whole thing a product, a quotient, or a composite? — and apply that rule, using the others inside it.

Where this goes wrong

  • The dropped chain factor — the single most common error in the topic. The power rule alone is only correct when the inside is a bare x, and here it runs three times as fast.

  • The inside was differentiated where it stood instead of being left alone and multiplied on the outside. The inside never changes inside the outer function.

  • Different functions, and their derivatives are nothing alike: against . Test at if it ever looks ambiguous.

  • The power rule applied to a constant base. An exponential never loses its exponent — it keeps it and picks up the derivative of it as a factor.

  • The inside stays inside the root; its derivative appears as a factor on top. Differentiating underneath the root is the same mistake as the sine one, wearing a different hat.

Practice
Name the outside and the inside before differentiating anything.

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