The Chain Rule as a structure-reading skill

See layers, not symbols

Ask what the last operation performed is. In sin(x²), squaring happens first and sine happens last, so sine is the outside layer and x² is the inside layer.

Peel one layer at a time. Deep compositions may require the rule repeatedly.

Why the inner derivative appears

The outer output changes according to its own slope, but its input is moving at the rate g′(x). Multiplying the rates accounts for both parts of the dependency.

Units tell the same story: output per inner unit times inner units per x-unit gives output per x-unit.

A dependable notation

Temporarily name the inside u when the structure is crowded. Differentiate with respect to u, then multiply by du/dx. The substitution is organizational, not a separate theorem.

Worked example

Common mistakes

• Differentiating the inside but forgetting the outside derivative.
• Replacing the inside with its derivative instead of multiplying.
• Stopping after one layer in a nested composition.

Keep these ideas

• Identify the last operation first.
• Keep the inside intact while differentiating the outside.
• Multiply by every inner rate encountered.