The inner derivative: the factor students leave behind, and the one the exam counts on

01The mistake

Students differentiate $\sin(3x)$ to $\cos(3x)$ instead of $3\cos(3x)$, and $(x^2+1)^5$ to $5(x^2+1)^4$ instead of $10x(x^2+1)^4$. The outer rule is applied correctly and the inner derivative is never brought down.

The tell is that the error is silent. $\cos(3x)$ is a perfectly well-formed expression that looks like a finished derivative, so nothing in the answer signals incompleteness. Compare this to a sign error or an algebra slip, which often produce something visibly odd. A student can check their work carefully and find nothing wrong.

It survives because it works when the inner derivative is 1. $\frac{d}{dx}\sin(x+2) = \cos(x+2)$ is correct, and so is every case where the inside is $x$ plus a constant. Students who practise on those cases get confirmation, and the rule they are confirming is the wrong one.

It compounds in Unit 3 as U3-CA2, where multi-layer compositions lose one factor out of three, and as U3-CA4, where implicit differentiation drops the $\frac{dy}{dx}$ on $y$-terms. That last one is the same error: $y$ is a function of $x$, so differentiating $y^2$ requires the chain rule, and the missing $\frac{dy}{dx}$ is the missing inner derivative.

02Why it makes sense to the student

The outer function is where the attention goes. Students see $\sin(\text{something})$, recall that sine differentiates to cosine, and produce it. The inner expression has already been processed as a unit — a thing sitting inside — rather than as a function that will itself need differentiating.

Nothing in the notation flags the composition. $\sin(3x)$ and $\sin(x)$ look almost identical, and the difference between them is a coefficient that appears to be part of the input rather than a separate function. There is no visual cue that two rules are in play.

Unit 2 trained single-rule differentiation exclusively. Power rule, product rule, quotient rule — each problem announced its own rule by its shape, and applying one rule was always the complete answer. The habit of asking “is there another layer” has not been built because it has never been needed.

And the answer looks right. This is the deepest reason it persists: there is no internal signal. A student with good habits who checks their work will confirm a wrong answer, because the check is against the outer rule, which was applied correctly.

03The correction

Force the decomposition before any differentiation. Write $u = $ the inside, then $f(u) = $ the outside, then differentiate both, then multiply. Making students name $u$ explicitly is what converts “a thing inside” into “a function that has a derivative,” and that reframing is the entire repair.

Give them the recognition question as a reflex: is this function applied to something other than plain $x$? If yes, the chain rule is required. Asking this before writing anything catches the cases where the composition is not obvious, which are the cases where it is missed.

Use the Leibniz form to make the missing factor structural rather than a rule to remember: $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$. The $du$ cancels, and the shape of the expression shows why a second factor must be there. Students who have seen it this way have a reason for the factor instead of a memory of it.

Then give them a check that actually detects the error, since their usual check cannot. Pick a value and compare against a numerical derivative: for $y=(x^2+1)^5$ at $x=1$, the correct derivative is $10 \cdot 1 \cdot 2^4 = 160$, while the incomplete version gives $5 \cdot 2^4 = 80$. A factor of two apart, and computable in a few seconds on a calculator.

A useful classroom test: put $\sin(x)$, $\sin(2x)$, $\sin(x+2)$ and $\sin(x^2)$ on the board and ask for all four derivatives. The first and third need no inner factor, the second and fourth do. A student who produces $\cos$ of the inside four times has the misconception; the two cases where they are accidentally right are what has been hiding it.

04A sample question

05What each wrong answer reveals

• A The inner derivative dropped. The power rule was applied correctly to the outer function and the $2x$ never appeared. This is the single most common error in the course, and the crucial point for teaching is that the student's work is not sloppy — it is complete by their own standard. Give them the numerical check (160 against 80 at $x=1$), because it is the only check that detects this.
• B Correct. With $u = x^2+1$, $\frac{dy}{du} = 5u^4$ and $\frac{du}{dx} = 2x$, so $\frac{dy}{dx} = 5(x^2+1)^4 \cdot 2x = 10x(x^2+1)^4$.
• C The inner derivative substituted for the inner function. The student differentiated the inside and then placed the result where the inside belonged, producing $5(2x)^4$. This is a different error from A — they know the inner derivative is needed, which is the conceptual step — and have used it in the wrong position. Explicit $u$-substitution notation fixes it quickly.
• D The outer power rule skipped. The inner derivative is present and the exponent was never reduced. Usually this is a student who has learned “multiply by the inside derivative” as a bolt-on step and applied it to the original expression rather than to the differentiated one. They have the factor and not the sequence.

A is the misconception; C and D are students who know a second factor belongs and have mis-sequenced it. That is a meaningful difference, because C and D will respond to writing the substitution out formally while A needs to be shown that an answer can look finished and be wrong. The four-way sine question above is worth running first, since it exposes A specifically.

06Try it in Mistake Master