Expanding each block
With the skeleton
\[f'(x)=\big[,\text{block for }u,\big]\cdot v+u\cdot\big[,\text{block for }v,\big]\]
in place, expand each block using the labeled-product layout.
Block for \(u=(x^2+1)^7\)
Outer function: \((;)^7\). Inner function: \(x^2+1\).
\[u'=7(x^2+1)^6\cdot 2x\]
The outer factor \(7(x^2+1)^6\) keeps the inner expression \(x^2+1\) copied unchanged. The inner factor is \(2x\).
Block for \(v=(3x-7)^4\)
Outer function: \((;)^4\). Inner function: \(3x-7\).
\[v'=4(3x-7)^3\cdot 3\]
Again the inner expression \(3x-7\) is copied unchanged into the outer factor, and the inner factor is the constant \(3\).
Inserting the blocks
\[f'(x)=\big7(x^2+1)^6\cdot 2x\big^4+(x^2+1)^7\big[4(3x-7)^3\cdot 3\big]\]
Each block sits in its own slot. The factor \(v\) is untouched in the first term; the factor \(u\) is untouched in the second term. Nothing has been simplified yet.
Why the blocks stay separated
If you simplify inside the first block before writing the second, you lose the visual cue that the first block belongs to \(u\) and the second belongs to \(v\). Keeping the blocks bracketed makes the audit on the next page possible: two composite factors, two chain rule blocks, four factors total in the first term and four in the second.
What this page establishes
The product rule template is filled with two expanded chain rule blocks, each in its own slot, with no factor dropped. Simplification is deliberately postponed so the structure remains visible.