The problem
Differentiate \(x^3+y^3=6xy\) with respect to \(x\).
Term-by-term rows
- \(\frac{d}{dx}\left[x^3\right]=3x^2\)
- \(\frac{d}{dx}\left[y^3\right]=3y^2\frac{dy}{dx}\)
- \(\frac{d}{dx}[6xy]\): the product rule gives \(6\left(1\cdot y+x\cdot\frac{dy}{dx}\right)=6y+6x\frac{dy}{dx}\)
Assembling:
\[3x^2+3y^2\frac{dy}{dx}=6y+6x\frac{dy}{dx}\]
Collecting:
\[3y^2\frac{dy}{dx}-6x\frac{dy}{dx}=6y-3x^2\]
\[\frac{dy}{dx}\left(3y^2-6x\right)=6y-3x^2\]
\[\frac{dy}{dx}=\frac{6y-3x^2}{3y^2-6x}=\frac{2y-x^2}{y^2-2x}\]
Check 1: layer count
Each differentiated term should produce as many factors as it has layers.
- \(x^3\): one layer, one factor, \(3x^2\). Correct.
- \(y^3\): two layers, two factors, \(3y^2\) and \(\frac{dy}{dx}\). Correct.
- \(6xy\): the product rule supplies two terms, and only the \(y\)-containing term carries a chain rule block. Correct.
Check 2: factor count
Every term whose innermost layer is \(y\) must carry exactly one \(\frac{dy}{dx}\) factor. In the assembled line, \(3y^2\frac{dy}{dx}\) and \(6x\frac{dy}{dx}\) each carry one, and no other term carries any. Correct.
Check 3: inner-function preservation
Inside each outer derivative factor, the inner expression must be copied unchanged.
- \(3y^2\) keeps \(y\) intact.
- \(6x\) is the derivative of the \(x\) factor, not an outer derivative, so no substitution applies.
Correct.
What a failure would look like
If the \(y^3\) term had been written as \(3y^2\) with no \(\frac{dy}{dx}\), check 2 would fail and name that term. If it had been written as \(3x^2\frac{dy}{dx}\), check 3 would fail, because the inner \(y\) was replaced by \(x\).
The audit as a habit
Run the three checks before simplifying. Each one is cheap, and each one targets a different way the layout can break.