The Rule Is Read Off the Root, Not Off the Symbols

You'll learnto name an expression's outermost operation before touching a rule, to rewrite when rewriting collapses the structure, and to see why a correct answer reached the long way still costs you marks you did not have to spend.

By this point every rule you need is already yours. What is left is the step before them: looking at an expression and naming its shape. Draw any expression as a tree and one node sits at the top — a sum, a product, a quotient, a composition — and that node alone picks the rule. Symbols lower down are loud and they are not the question. Most lost marks in this topic are not failed computations. They are correct computations of the wrong kind, or heroic effort that one line of algebra would have deleted.

8 STEPS · 6 QUICK CHECKS · NAME THE ROOT · REWRITE · THEN DIFFERENTIATE · v1

root ⇄ rule · rewrite ⇄ a smaller tree

Before you start

What you're looking at

Expressions drawn as trees. Every one has a single node at the top, and that node — not the symbols you happen to notice — is what names the rule.

The question

Two students get the same right answer, one in two lines and one in seven. What did the first one see that the second one did not?

Watch for

Step 6, where expanding a logarithm collapses a four-deep tree into two easy terms — and step 7, where the line counter, not a red cross, is what convicts the expensive route.

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