Step by Step
1
What a composite function is
A composite function is a function inside another function, like sin(x²) — here, the "outer" function is sin( ) and the "inner" function is x².
2
The rule itself
d/dx[f(g(x))] = f'(g(x)) · g'(x) — take the derivative of the outer function, leaving the inner function completely untouched inside it, then multiply by the derivative of the inner function.
3
The step everyone forgets
Multiplying by the inner derivative (g') at the end is the single most frequently skipped step. Forgetting it is the most common error on any chain rule problem.
4
Chains can nest multiple times
Some functions have more than two layers — like sin((x²+1)³). In these cases, keep applying the chain rule outward to inward, multiplying by each additional layer's derivative as you go.
Applied Walkthrough
1
Differentiate f(x) = sin(x²). The outer function is sin( ), and the inner function is x².
2
Differentiate the outer function while leaving the inside untouched: cos(x²).
3
Multiply by the derivative of the inner function: d/dx[x²] = 2x.
4
Combine: f'(x) = cos(x²) · 2x = 2x·cos(x²). Forgetting to multiply by 2x here is the most common chain rule mistake.
Exam Application
Exams test whether you correctly identify the outer and inner functions, and specifically whether you remember to multiply by the inner function's derivative — the step most frequently skipped.
⚠ Common Trap
The single biggest trap is stopping after differentiating just the outer function and forgetting to multiply by the inner derivative. cos(x²) alone is not the correct derivative of sin(x²) — you must also multiply by 2x.
✓ Quick Self-Check
1. What is the chain rule formula?
d/dx[f(g(x))] = f'(g(x)) · g'(x).
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2. Differentiate f(x) = sin(x²).
f'(x) = 2x·cos(x²).
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3. What is the most commonly skipped step in the chain rule?
Multiplying by the derivative of the inner function.
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4. In f(g(x)), which part is left unchanged when you differentiate the outer function?
The inner function, g(x) — it stays exactly as it is inside the outer function's derivative.
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5. How do you handle a function with more than two nested layers?
Keep applying the chain rule outward to inward, multiplying by each additional layer's derivative in turn.
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