The Chain Rule
- Differentiate compositions of functions with the chain rule
- Identify the outer and inner functions in a composition
- Combine the chain rule with the product and quotient rules
Composition demands the chain rule
When one function is nested inside another — f(g(x)) — differentiating requires the chain rule: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function. Symbolically, the rate of the whole is the rate of the outer times the rate of the inner. Miss the inner derivative and you have only done half the job; that missing factor is the single most common differentiation mistake in the course.
Peeling the layers
The skill is spotting the layers. In sin(x²), the outer is sine and the inner is x². In (3x + 1)⁵, the outer is "raise to the fifth" and the inner is 3x + 1. Work from the outside in: differentiate the outermost operation first, copy the inside untouched, then multiply by the derivative of what is inside. For deeply nested functions you simply keep applying the rule, multiplying one inner-derivative factor per layer.
Differentiate y = (3x² + 1)⁴.
- 1.Identify layers: outer is u⁴ (raise to the fourth), inner is u = 3x² + 1.
- 2.Differentiate the outer, keeping the inside: d/du[u⁴] = 4u³ → 4(3x² + 1)³.
- 3.Differentiate the inner: d/dx[3x² + 1] = 6x.
- 4.Multiply the two factors: y′ = 4(3x² + 1)³ · 6x = 24x(3x² + 1)³.
After differentiating the outer function, ask yourself: "times the derivative of the inside?" Making that question a reflex guarantees you never drop the g′(x) factor.
What is the derivative of y = sin(x²)?
What is the derivative of y = e^(5x)?
Chain rule questions escalate by nesting. For sin³(2x) = (sin(2x))³ you apply the rule twice: 3(sin 2x)² · cos(2x) · 2. Count the layers before you start so you know how many inner-derivative factors to expect.
Answer the 2 checkpoints as you read.
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