The Chain Rule as a structure-reading skill
Differentiate the outside function, keep the inside intact, then multiply by the inside rate.
How to recognize the method, run it, and know when it is the wrong choice.
Reviewed July 11, 2026Use the Chain Rule whenever one varying expression has been substituted inside another function.
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.
Differentiate (1 + e^(3x))⁵.
Differentiate the fifth-power layer.
Differentiate the 1 + exponential layer.
Differentiate the exponent 3x.
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.
Three takeaways
- Identify the last operation first.
- Keep the inside intact while differentiating the outside.
- Multiply by every inner rate encountered.